Let’s consider E to be the event of getting even number on tossing a die.
A die is tossed twice. A ‘success’ is getting an even number on a toss. Find the variance of the number of successes.
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Probability
Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces.
Let’s consider X to be the random variable such that X = \[0,1,2\] Now, let E = the event of drawing an ace And, F = the event of drawing non – ace So,
A bag contains \[(2n+1)\] coins. It is known that n of these coins has a head on both sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is \[31/42\], determine the value of n.
Given, n coins are two headed coins and the remaining \[(n+1)\] coins are fair. Let \[{{E}_{1}}\] : the event that unfair coin is selected \[{{E}_{2}}\] : the event that the fair coin is selected...
The probability distribution of a random variable x is given as under: where k is a constant. Calculate (i) E(X) (ii) \[\mathbf{E}\text{ }(\mathbf{3}{{\mathbf{X}}^{\mathbf{2}}})\] (iii) \[\mathbf{P}\left( \mathbf{X}\text{ }{}^\text{3}\text{ }\mathbf{4} \right)\]
The probability distribution of random variable X is given by: We know that \[\sum\limits_{i=1}^{n}{P({{X}_{i}})=1}\] So, \[k\text{ }+\text{ }4k\text{ }+\text{ }9k\text{ }+\text{ }8k\text{ }+\text{...
The probability distribution of a discrete random variable X is given as under: Calculate: (i) The value of A if E(X) = \[2.94\] (ii) Variance of X.
(i) We know that: (ii) Now, the distribution becomes \[E({{X}^{2}})\text{ }=\text{ }1\text{ }\times \text{ 1/2 }+\text{ }4\text{ }\times \text{ }1/5\text{ }+\text{ }16\text{ }\times 3/25\text{...
Let X be a discrete random variable whose probability distribution is defined as follows: where k is a constant. Calculate (i) the value of k (ii) E (X) (iii) Standard deviation of X.
(i) Given, \[P\left( X\text{ }=\text{ }x \right)\text{ }=\text{ }k\left( x\text{ }+\text{ }1 \right)\]for \[x\text{ }=\text{ }1,\text{ }2,\text{ }3,\text{ }4\] So, \[P\left( X\text{ }=\text{ }1...
An item is manufactured by three machines A, B and C. Out of the total number of items manufactured during a specified period, \[50%\] are manufactured on A, \[30%\] on B and \[20%\] on C. \[2%\] of the items produced on A and \[2%\] of items produced on B are defective, and \[3%\] of these produced on C are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
Let’s consider: \[{{E}_{1}}\] = The event that the item is manufactured on machine A \[{{E}_{2}}\] = The event that the item is manufactured on machine B \[{{E}_{3}}\] = The event that the item is...
By examining the chest X ray, the probability that TB is detected when a person is actually suffering is \[0.99\]. The probability of an healthy person diagnosed to have TB is \[0.001\]. In a certain city, \[1\] in \[1000\] people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?
Let \[{{E}_{1}}\] = Event that a person has TB \[{{E}_{2}}\] = Event that a person does not have TB And H = Event that the person is diagnosed to have TB. So, \[P({{E}_{1}})\text{ }=\text{...
There are three urns containing \[2\]white and \[3\] black balls, \[3\] white and \[2\] black balls, and \[4\] white and \[1\] black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.
Given, we have \[3\] urns: Urn \[1\] = \[2\] white and \[3\] black balls Urn \[2\] = \[3\] white and 2 black balls Urn \[3\] = \[4\] white and \[1\] black balls Now, the probabilities of choosing...
There are two bags, one of which contains \[3\] black and \[4\] white balls while the other contains \[4\] black and \[3\] white balls. A die is thrown. If it shows up \[1\] or \[3\], a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.
Let \[{{E}_{1}}\] be the event of selecting Bag \[1\] and \[{{E}_{2}}\] be the event of selecting Bag \[2\]. Also, let \[{{E}_{3}}\] be the event that black ball is selected Now,...
A shopkeeper sells three types of flower seeds \[{{A}_{1}}\], \[{{A}_{2}}\] and \[{{A}_{3}}\]. They are sold as a mixture where the proportions are \[4:4:2\] respectively. The germination rates of the three types of seeds are \[45%\], \[60%\] and \[35%\]. Calculate the probability (i) of a randomly chosen seed to germinate (ii) that it will not germinate given that the seed is of type A3, (iii) that it is of the type A2 given that a randomly chosen seed does not germinate.
Given that: \[{{A}_{1}}:\text{ }{{A}_{2}}:\text{ }{{A}_{3}}~=\text{ }4:\text{ }4:\text{ }2\] So, the probabilities will be \[P({{A}_{1}})\text{ }=\text{ }4/10,\text{ }P({{A}_{2}})\text{ }=\text{...
Refer to Question 41 above. If a white ball is selected, what is the probability that it came from (i) Bag \[2\] (ii) Bag \[3\]
Referring the Question 41, here we will use Baye’s Theorem Therefore, the required probabilities are \[2/11\] and \[9/11\].
Three bags contain a number of red and white balls as follows: Bag \[1:3\] red balls, Bag \[2:2\] red balls and \[1\] white ball Bag \[3:3\] white balls. The probability that bag i will be chosen and a ball is selected from it is \[\mathbf{i}/\mathbf{6},\text{ }\mathbf{i}\text{ }=\text{ }\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3}\]. What is the probability that (i) a red ball will be selected? (ii) a white ball is selected?
Given: Bag \[1:3\] red balls, Bag \[2:2\] red balls and \[1\] white ball Bag \[3:3\] white balls Now, let E1, E2 and E3 be the events of choosing Bag \[1\], Bag \[2\] and Bag \[3\] respectively and...
An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.
Let’s consider A to be the event of having m white and n black balls \[{{E}_{1}}\] = First ball drawn of white colour \[{{E}_{2}}\] = First ball drawn of black colour \[{{E}_{3}}\] = Second ball...
Two dice are tossed. Find whether the following two events A and B are independent: \[\mathbf{A}\text{ }=\text{ }\left\{ \left( x,~y \right)\text{ }:~x~+~y~=\text{ }\mathbf{11} \right\}\text{ }\mathbf{B}\text{ }=\text{ }\left\{ \left( x,~y \right)\text{ }:~x~{}^\text{1}\text{ }\mathbf{5} \right\}\] where (x, y) denotes a typical sample point.
Given, two events A and B are independent such that \[\mathbf{A}\text{ }=\text{ }\left\{ \left( x,~y \right)\text{ }:~x~+~y~=\text{ }\mathbf{11} \right\}\text{ }\mathbf{B}\text{ }=\text{ }\left\{...
A and B throw a pair of dice alternately. A wins the game if he gets a total of \[6\] and B wins if she gets a total of \[7\]. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.
Solution: Let’s take A1 to be the event of getting a total of \[6\] \[{{A}_{1}}~=\text{ }\left\{ \left( 2,\text{ }4 \right),\text{ }\left( 4,\text{ }2 \right),\text{ }\left( 1,\text{ }5...
Find the variance of the distribution:
We know that, Variance(X) = \[E({{X}^{2}})\text{ }\text{ }{{\left[ E\left( X \right) \right]}^{2}}\] Therefore, the required variance is \[665/324\].
The random variable X can take only the values \[0,1,2\]. Given that \[\mathbf{P}\left( \mathbf{X}\text{ }=\text{ }\mathbf{0} \right)\text{ }=\text{ }\mathbf{P}\text{ }\left( \mathbf{X}\text{ }=\text{ }\mathbf{1} \right)\text{ }=~p~\] and that \[\mathbf{E}({{\mathbf{X}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{E}\left[ \mathbf{X} \right]\],find the value of p.
Given, \[X\text{ }=\text{ }0,\text{ }1,\text{ }2\]and \[P\left( X\text{ }=\text{ }0 \right)\text{ }=\text{ }P\text{ }\left( X\text{ }=\text{ }1 \right)\text{ }=~p\] Let P(X) at \[X\text{ }=\text{...
Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.
Let X be the random variable scores when a die is thrown twice. \[X\text{ }=\text{ }1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6\] And \[S\text{ }=\text{ }\left\{ \left( 1,\text{ }1...
Two natural numbers r, s are drawn one at a time, without replacement from the set \[\mathbf{S}=\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots .,~n \right\}\]. Find \[\mathbf{P}\left[ r~\text{£}~p|s~\text{£}~p \right]\], where \[p\mathbf{\hat{I}}S\].
Given, \[\mathbf{S}=\left\{ \mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\ldots .,~n \right\}\] So, \[P\left( r\text{ }\le \text{ }p/s\text{ }\le \text{ }p \right)\text{ }=\text{...
Suppose that \[6%\] of the people with blood group O are left-handed and \[10%\] of those with other blood groups are left-handed \[30%\] of the people have blood group O. If a left-handed person is selected at random, what is the probability that he/she will have blood group O?
Let’s assume \[{{E}_{1}}\] = The event that a person selected is of blood group O \[{{E}_{2}}\] = The event that the people selected is of other group And H = The event that selected person is left...
Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is \[2\]-headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?
Let’s consider \[{{E}_{1}}\] = Event that the coin is fair \[{{E}_{2}}\] = Event that the coin is \[2\] headed And H = Event that the tossed coin gets head. Now, \[P({{E}_{1}})\text{ }=\text{...
A factory produces bulbs. The probability that any one bulb is defective is \[1/50\] and they are packed in boxes of \[10\]. From a single box, find the probability that (i) none of the bulbs is defective (ii) exactly two bulbs are defective (iii) more than \[8\] bulbs work properly
Let’s assume X to be the random variable denoting a bulb to be defective. Here, \[n\text{ }=\text{ }10,\text{ }p\text{ }=\text{ }1/50,\text{ }q\text{ }=\text{ }1\text{ }\text{ }1/50\text{ }=\text{...
Two probability distributions of the discrete random variable X and Y are given below. Prove that \[\mathbf{E}({{\mathbf{Y}}^{\mathbf{2}}})\text{ }=\text{ }\mathbf{2}\text{ }\mathbf{E}\left( \mathbf{X} \right)\].
The probability distribution of random variable X is We know that, \[E(X)=\sum\limits_{i=1}^{n}{{{P}_{i}}{{X}_{i}}}\] \[E\left( X \right)\text{ }0.\text{ }1/5\text{ }+\text{ }1.\text{ }2/5\text{...
Two biased dice are thrown together. For the first die \[\mathbf{P}\left( \mathbf{6} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{2}\], the other scores being equally likely while for the second die, \[\mathbf{P}\left( \mathbf{1} \right)\text{ }=\text{ }\mathbf{2}/\mathbf{5}\]and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.
Therefore, the required probability distribution is
A die is thrown three times. Let X be ‘the number of twos seen’. Find the expectation of X.
Here, we have \[X\text{ }=\text{ }0,\text{ }1,\text{ }2,\text{ }3\] [Since, die is thrown \[3\] times] And \[p\text{ }=\text{ }1/6,\text{ }q\text{ }=\text{ }5/6\] Therefore, the required expression...
A biased die is such that \[\mathbf{P}\left( \mathbf{4} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{10}\]and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’, find the variance of the random variable X.
Here, random variable \[X\text{ }=\text{ }0,\text{ }1,\text{ }2\] \[P\left( X\text{ }=\text{ }2 \right)\text{ }=\text{ }P\left( 4 \right).P\left( 4 \right)\text{ }=\text{ }1/10\text{ }x\text{...
For the following probability distribution determine standard deviation of the random variable X.
We know that: Standard deviation (S.D.) = \[\sqrt{Variance}\] So, \[Var\left( X \right)\text{ }=\text{ }E({{X}^{2}})\text{ }\text{ }{{\left[ E\left( X \right) \right]}^{2}}\] \[E\left( X...
The probability distribution of a random variable X is given below: (i) Determine the value of k. (ii) Determine \[\mathbf{P}\left( \mathbf{X}\text{ }\text{£}\text{ }\mathbf{2} \right)\] and \[\mathbf{P}\left( \mathbf{X}\text{ }>\text{ }\mathbf{2} \right)\] (iii) Find \[\mathbf{P}\left( \mathbf{X}\text{ }\text{£}\text{ }\mathbf{2} \right)\text{ }+\text{ }\mathbf{P}\text{ }\left( \mathbf{X}\text{ }>\text{ }\mathbf{2} \right)\]
(i) W.k.t \[P\left( 0 \right)\text{ }+\text{ }P\left( 1 \right)\text{ }+\text{ }P\left( 2 \right)\text{ }+\text{ }P\left( 3 \right)\text{ }=\text{ }1\] \[\Rightarrow k\text{ }+\text{ }k/2\text{...
Consider the probability distribution of a random variable X: Calculate: (i) \[\mathbf{V}\left( \mathbf{X}/\mathbf{2} \right)\] (ii) Variance of X.
Given We know that: \[\mathbf{Var}\left( \mathbf{X} \right)\text{ }=\text{ }\mathbf{E}({{\mathbf{X}}^{\mathbf{2}}})\text{ }\text{ }{{\left[ \mathbf{E}\left( \mathbf{X} \right)...
A lot of \[100\] watches is known to have \[10\] defective watches. If 8 watches are selected (one by one with replacement) at random, what is the probability that there will be at least one defective watch?
Given: Total number of watches = \[100\] and number of defective watches = \[10\] So, the probability of selecting a detective watch = \[10/100\text{ }=\text{ }1/10\] Now, n = \[8\], \[p\text{...
The probability of a man hitting a target is \[0.25\]. He shoots \[7\] times. What is the probability of his hitting at least twice?
Here, we have n = \[7\], \[p\text{ }=\text{ }0.25\text{ }=\text{ }25/100\text{ }=\text{ 1/4}\] and \[q=1-1/4=3/4\] \[P\left( X\text{ }\ge \text{ }2 \right)\text{ }=\text{ }1\text{ }\text{ }\left[...
Ten coins are tossed. What is the probability of getting at least 8 heads?
Here we have, \[n=10\], \[p=1/2\] and \[q=1-1/2=1/2\] \[P\left( X\text{ }\ge \text{ }8 \right)\text{ }=\text{ }P\left( x\text{ }=\text{ }8 \right)\text{ }+\text{ }P\left( x\text{ }=\text{ }9...
A die is thrown \[5\] times. Find the probability that an odd number will come up exactly three times.
Here, \[p\text{ }=\text{ }1/6\text{ }+\text{ }1/6\text{ }+\text{ }1/6\text{ }=\text{ 1/2}\Rightarrow q\text{ }=\text{ }1\text{ }\text{ 1/2 }=\text{ 1/2}\] and \[n=5\] Now, \[P\left( x\text{ }=\text{...
Four cards are successively drawn without replacement from a deck of \[52\] playing cards. What is the probability that all the four cards are kings?
Let \[{{E}_{1}},{{E}_{2}},{{E}_{3}}\] and \[{{E}_{4}}\] be the events that first, second, third and fourth card is King respectively. Therefore, the required probability is \[1/27075\].
A box has \[5\] blue and \[4\] red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?
Given that the box has \[5\] blue and \[4\] red balls. Let us consider \[{{E}_{1}}\] be the event that first ball drawn is blue and \[{{E}_{2}}\] be the event that first ball drawn is red. And, E is...
Bag I contains \[3\] black and \[2\] white balls, Bag II contains \[2\] black and \[4\] white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.
According to the question: Bag \[1\] has \[3\]B, \[2\]W balls and Bag \[2\] has \[2\]B, \[4\]W balls. Let \[{{E}_{1}}\] = The event that bag \[1\] is selected \[{{E}_{2}}\] = The event that bag...
A bag contains \[4\] white and \[5\] black balls. Another bag contains \[9\] white and \[7\] black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.
Let us consider \[{{W}_{1}}\] and \[{{W}_{2}}\] to be two bags containing \[\left( 4W,\text{ }5B \right)\]and \[\left( 9W,\text{ }7B \right)\]balls respectively. Let us take \[{{E}_{1}}\] be the...
Suppose \[10,000\] tickets are sold in a lottery each for Re \[1\]. First prize is of Rs \[3000\] and the second prize is of Rs. \[2000\]. There are three third prizes of Rs. \[500\] each. If you buy one ticket, what is your expectation.
Let’s take X to be the random variable where X = \[0,500,2000\]and \[3000\]
Three dice are thrown at the same time. Find the probability of getting three two’s, if it is known that the sum of the numbers on the dice was six.
Given that the dice is thrown three times So, the sample space n(S) = \[{{6}^{3}}~=\text{ }216\] Let E1 be the event when the sum of number on the dice was \[6\] and \[{{E}_{2}}\]be the event when...
In a dice game, a player pays a stake of Re\[1\] for each throw of a die. She receives Rs \[5\] if the die shows a \[3\], Rs \[2\] if the die shows a \[1\]or \[6\], and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?
Let’s take X to be the random variable of profit per throw. As, she loses Rs \[1\] for giving any od \[2,4,5\]. So, \[P\left( X\text{ }=\text{ }-1 \right)\text{ }=\text{ }1/6\text{ }+\text{...
If X is the number of tails in three tosses of a coin, determine the standard deviation of X.
Given, \[X\text{ }=\text{ }0,\text{ }1,\text{ }2,\text{ }3\] P(X = r) \[={{~}^{n}}{{C}_{r}}~{{p}^{r}}~{{q}^{n-r}}\] Where \[n\text{ }=\text{ }3,\text{ }p\text{ }=\text{ 1/2},\text{ }q\text{ }=\text{...
Prove that (i) P(A) = \[\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{B} \right)\text{ }+\text{ }\mathbf{P}\left( \mathbf{A}~B~ \right)\] (ii) \[\mathbf{P}\left( \mathbf{A\grave{E}}\text{ }\mathbf{B} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{B} \right)\text{ }+\text{ }\mathbf{P}\left( \mathbf{A}~B~ \right)\text{ }+\text{ }\mathbf{P}\left( ~A~\mathbf{}\text{ }\mathbf{B} \right)\]
A discrete random variable X has the probability distribution given as below: (i) Find the value of k (ii) Determine the mean of the distribution.
For a probability distribution, we know that if \[{{P}_{i}}~\ge \text{ }0\]
Let E1 and E2 be two independent events such that \[p({{\mathbf{E}}_{\mathbf{1}}})\text{ }=~{{p}_{\mathbf{1}}}~\] and \[\mathbf{P}({{\mathbf{E}}_{\mathbf{2}}})\text{ }=\text{ }{{\mathbf{p}}_{\mathbf{2}}}\]. Describe in words of the events whose probabilities are: \[\left( \mathbf{i} \right)~{{p}_{\mathbf{1}}}~{{p}_{\mathbf{2}}}~\left( \mathbf{ii} \right)\text{ }(\mathbf{1}{{p}_{\mathbf{1}}})~{{p}_{\mathbf{2}}}~\left( \mathbf{iii} \right)\text{ }\mathbf{1}\text{ }\text{ }(\mathbf{1}\text{ }~{{p}_{\mathbf{1}}})(\mathbf{1}\text{ }~{{p}_{\mathbf{2}}})\text{ }\left( \mathbf{iv} \right)~{{p}_{\mathbf{1}}}~+~{{p}_{\mathbf{2}}}~\text{ }\mathbf{2}{{p}_{\mathbf{1}}}{{p}_{\mathbf{2}}}\]
Here, \[p({{\mathbf{E}}_{\mathbf{1}}})\text{ }=~{{p}_{\mathbf{1}}}~\] and \[\mathbf{P}({{\mathbf{E}}_{\mathbf{2}}})\text{ }=\text{ }{{\mathbf{p}}_{\mathbf{2}}}\] Now, its clearly seen that either...
Three events A, B and C have probabilities \[2/5\], \[1/3\] and \[1/2\] respectively. Given that \[\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{C} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{5}\] and \[\mathbf{P}\left( \mathbf{B}\text{ }\mathbf{}\text{ }\mathbf{C} \right)\text{ }=\text{ 1/4}\], find the values of \[\mathbf{P}\left( \mathbf{C}\text{ }|\text{ }\mathbf{B} \right)\] and \[\mathbf{P}\left( \mathbf{A}\mathbf{}\text{ }\mathbf{C} \right)\].
Given, P(A) = \[2/5\], P(B) = \[1/3\] and P(C) = \[1/2\] \[\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{C} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{5}\]and \[\mathbf{P}\left( \mathbf{B}\text{...
A and B are two events such that P(A) = \[1/2\], P(B) = \[1/3\] and \[\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{}\text{ }\mathbf{B} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{4}\]. Find: (i) \[\mathbf{P}\left( \mathbf{A}|\mathbf{B} \right)\] (ii) \[\mathbf{P}\left( \mathbf{B}|\mathbf{A} \right)\] (iii) \[\mathbf{P}\left( \mathbf{A}|\mathbf{B} \right)\] (iv) \[\mathbf{P}\left( \mathbf{A}|\mathbf{B} \right)\]
Given, P(A) = \[1/2\], P(B) = \[1/3\] and \[\mathbf{P}\left( \mathbf{A}\text{ }\mathbf{}\text{ }\mathbf{B} \right)\text{ }=\text{ }\mathbf{1}/\mathbf{4}\] \[P\left( A \right)\text{ }=\text{ }1\text{...
Explain why the experiment of tossing a coin three times is said to have binomial distribution.
As the random variable X takes \[0,1,2,3,......n\] is said to be binomial distribution having parameters n and p, if the probability is given by P(X = r) = \[^{n}{{C}_{r~}}{{p}^{r~}}{{q}^{n-r}}\]...
Two dice are thrown together and the total score is noted. The events E, F and G are ‘a total of \[4\]’, ‘a total of \[9\] or more’, and ‘a total divisible by \[5\]’, respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.
If two dice are thrown together, we have n(S) = \[36\] Now, let’s consider: E = A total of \[4\text{ }=\text{ }\left\{ \left( 2,\text{ }2 \right),\text{ }\left( 1,\text{ }3 \right),\text{ }\left(...
A bag contains \[5\] red marbles and \[3\] black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?
Let red marbles be presented with R and black marble with B. Also, let E be the event that at least one of the three marbles drawn be black when the first marble is red. Now, the following three...
The probability that at least one of the two events A and B occurs is \[0.6\]. If A and B occur simultaneously with probability \[0.3\], evaluate \[P(\overline{A})+P(\overline{B})\]
W.k.t, \[A\cup B\] denotes that atleast one of the events occurs and \[A\cap B\] denotes that two events occur simultaneously. Therefore, the required answer is \[1.1\].
Refer to Exercise 1 above. If the die were fair, determine whether or not the events A and B are independent.
According to the solution of exercise 1, we have \[A\text{ }=\text{ }\left\{ \left( 1,\text{ }1 \right),\text{ }\left( 2,\text{ }2 \right),\text{ }\left( 3,\text{ }3 \right),\text{ }\left( 4,\text{...
For a loaded die, the probabilities of outcomes are given as under: \[\mathbf{P}\left( \mathbf{1} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{2} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{2},\text{ }\mathbf{P}\left( \mathbf{3} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{5} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{6} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{1}\text{ }\mathbf{and}\text{ }\mathbf{P}\left( \mathbf{4} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{3}\]. The die is thrown two times. Let A and B be the events, ‘same number each time’, and ‘a total score is \[10\] or more’, respectively. Determine whether or not A and B are independent.
Given that a loaded die is thrown such that \[\mathbf{P}\left( \mathbf{1} \right)\text{ }=\text{ }\mathbf{P}\left( \mathbf{2} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{2},\text{ }\mathbf{P}\left(...