Probability

### Two dice are rolled together. Find the probability o f getting such numbers on two dice whose product Is perfect square.

Solution: When two different dice are thrown, then total number of outcomes = 36. Let E be the event of getting the product of numbers, as a perfect square. These numbers are (1,1), (1,4), (2,2),...

### When two dice are tossed together, find the probability that the sum o f the numbers on their tops Is less than 7.

Solution: When two different dice are thrown, the total number of outcomes = 36. Let E be the event of getting the sum of the numbers less than 7. These numbers are (1,1), (1,2), (1,3), (1,4),...

### Two different dice are rolled simultaneously. Find the probability that the sum o f the numbers on the two dice Is 10.

Solution: When two different dice are thrown, the total number of outcomes = 36.   Let E1 be the event of getting the sum of the numbers on the two dice is 10. These numbers are (4 ,6), (5,5)...

### A coin is tossed. If it shows tail, we draw a ball from a box which contains 2 red 3 black balls; it shows head, we throw a die. Find the sample space of this experiment.

According to the question, A coin is tossed and there is box which contains 2 red and 3 black balls. When coin is tossed, the outcomes will be {H, T} According to question, if tail is turned up,...

### An experiment consists of tossing a coin and then tossing it second time if head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.

According to the question, a coin is tossed and if the outcome is tail then, a die will be rolled. The possible outcome for coin is 2 that is {H, T} And, the possible outcome for die is 6 that is...

### A coin is tossed and then a die is rolled only in case a head is shown on the coin. Describe the sample space for this experiment.

According to the question, a coin is tossed and the die is rolled. So, when coin is tossed there will be 2 events that is either Head or tail, According to question, If Head occurs on coin then Die...

### What is the total number of elementary events associated to the random experiment of throwing three dice together?

According to the question, three dice are thrown together. So there are 6 faces on die. As a result, the total numbers of elementary events on throwing three dice are $6^3=216$

### Two dice are thrown. Describe the sample space of this experiment.

As we know, there are 6 faces on a dice containing (1, 2, 3, 4, 5, 6). According to the question, two dice are thrown, so we have two faces of dice (one of each). As a result, the total sample space...

### Write the sample space for the experiment of tossing a coin four times.

According to the question, a coin is tossed four time, so the no. of samples will be, $2^4=16$ So, S = {HHHH, TTTT, HHHT, HHTH, HTHH, THHH, HHTT, HTTH, HTHT, THHT, THTH, TTHH, HTTT, THTT, TTHT,...

### If a coin is tossed two times, describe the sample space associated to this experiment.

We know that if two coins are tossed, that means two probabilities will occur at same time. So, S = {HT, TH, HH, TT} ∴ Sample space is {HT, HH, TT, TH}

### Choose the correct answer out of four given options in each of the exercise without repetition of the numbers, four digit numbers are formed with the numbers The probability of such a number divisible by 5 is A. B. C. D.

Solution: Option (D) $5 / 9$ Explanation: The digits $0,2,3,5$ we have. It is known that, if unit place digit is ${ }^{\prime} 0^{\prime}$ or ${ }^{\prime} 5^{\prime}$ then the no. is divisible by...

### Choose the correct answer out of four given options in each of the exercise seven persons are to be seated in a row. The probability that two particular persons sit next to each other is A. B. C. D.

Solution: Option (C) $2 / 7$ is correct. Explanation: It is given that 7 persons are to be seated in a row. If two persons sit next to each other, then they'll be considered as 1 group. We now have...

### Choose the correct answer out of four given options in each of the exercise while shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours A. B. C. D.

Solution: Option (C) $26 / 51$ is correct. Explanation: It is known that, in a deck of 52 cards 26 are red and 26 are of black. Given that 2 cards are accidentally dropped Therefore, Probability of...

### Choose the correct answer out of four given options in each of the exercise three numbers are chosen from 1 to 20 . Find the probability that they are not consecutive A. B. C. D.

Solution: Option (B) $187 / 190$ is correct. Explanation: As, the set of 3 consecutive nos. from 1 to 20 are $(1,2,3),(2,3,4),(3,4,5), \ldots,(18,19,20)$ Consider the 3 numbers as a single digit...

### One urn contains two black balls (labelled and ) and one white ball. A second urn contains one black ball and two white balls (labelled and ). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. (a) Write the sample space showing all possible outcomes (b) What is the probability that two black balls are chosen?

Solution: It is given that one urn contains two black balls and one white ball and the second urn contains one black ball and two white balls. Also given that if one of the two urns is chosen, then...

### The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P (A ∩ B) = .07. Determine (a) P (B ∩ C) (b) Probability of exactly one of the three occurs.

Solution: It is given that $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.07$ From the Venn Diagram given (a) $P(B \cap C)$ From the Venn diagram we have $P(B \cap C)=0.15$ (b) Probability of exactly one...

### The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P (A ∩ B) = .07. Determine (a) P (A ∪ B) (b) P (A ∩ B)

Solution: It is given that $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.07$ From the Venn Diagram given (a) $P(A \cup B)$ Using the General Addition Rule, $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ Substitute...

### The accompanying Venn diagram shows three events, A, B, and C, and also the probabilities of the various intersections (for instance, P (A ∩ B) = .07. Determine (a) P (A) (b)

Solution: It is given that $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.07$ From the Venn Diagram given (a) P(A) (b) $P(B \cap \bar{C})$ $P(B \cap \bar{C})=P(B)-P(B \cap C)$ Substitute the values,...

### One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently, the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of John’s promotion is same as that of Gurpreet, Rita’s chances of promotion are twice as likely as Johns. Aslam’s chances are four times that of John. (a) Determine P (John promoted) P (Rita promoted) P (Aslam promoted) P (Gurpreet promoted) (b) If A = {John promoted or Gurpreet promoted}, find P (A).

Solution: Given that: Sample Space, S $=$ John promoted, Rita promoted, Aslam promoted, Gurpreet promoted Let's say Events that John promoted $E_{1}$ Events that Rita promoted $E_2$ Events that...

### Four candidates A, B, C, D have applied for the assignment to coach a school cricket team. If A is nd C are given about the same chance of being selected, D, what are the probabilities that (a) C will be selected? (b) A will not be selected?

Solution: It is given that A is twice as likely to be selected as $B$ that is $P(A)=2 P(B) \ldots \ldots 1$ and $C$ is twice as likely to be selected as $D$. that is $P(C)=2 P(D) \ldots \ldots 2$ So...

### A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, . Find the probabilities that a particular surgery will be rated. (a) routine or complex (b) routine or simple

Solution: Let' say Event that surgeries are rated as very complex $=E_{1}$ Event that surgeries are rated as complex $=\mathrm{E}_{2}$ Event that surgeries are rated as routine $=\mathrm{E}_{3}$...

### A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, . Find the probabilities that a particular surgery will be rated. (a) complex or very complex; (b) neither very complex nor very simple;

Solution: Let' say Event that surgeries are rated as very complex $=E_{1}$ Event that surgeries are rated as complex $=\mathrm{E}_{2}$ Event that surgeries are rated as routine $=\mathrm{E}_{3}$...

### If A and B are mutually exclusive events, P (A) = 0.35 and P (B) = 0.45, find (a) P (A′) (b) P (B′)

Solution: Given: $P(A)=0.35$ and $P(B)=0.45$ $\because$ The events $A$ and $B$ are mutually exclusive then $P(A \cap B)=0$ (a) We need to find $P\left(A^{\prime}\right)$ It is known that,...

### In a large metropolitan area, the probabilities are .87, .36, .30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either anyone or both kinds of sets?

Solution: Event that a family owns colour television $=E_{1}$ Event that the family owns black and white television $=E_{2}$ Provided that $P\left(E_{1}\right)=0.87$ $P\left(E_{2}\right)=0.36$ and...

### A container contains 24 marbles, some are green and others are blue. In the event that a marble is drawn at arbitrary from the container, the likelihood that it is green is ⅔. Track down the quantity of blue balls n the container.

Solution: Absolute marbles = 24 Let the absolute green marbles = x Along these lines, the complete blue marbles = 24-x P(getting green marble) = x/24 From the inquiry, x/24 = ⅔ Along these lines,...

### A crate contains 12 balls out of which x are dark. On the off chance that one ball is drawn indiscriminately from thebox, what is the likelihood that it will be a debase?On the off chance that 6 more torpedoes are placed in the case, the likelihood of drawing a renounce is presentlytwofold of what it was previously. Discover x

Solution: All out number of debases = x All out number of balls = 12 P(E) = (Number of positive results/Total number of results) P (getting debases) = x/12 — — — - (I) Presently, when 6 more debases...

### A sack contains 5 red balls and some blue balls. In the event that the likelihood of drawing a blue ball is twofold that of a red ball, decide the quantity of blue balls taken care of.

Solution: It is given that the absolute number of red balls = 5 Let the all out number of blue balls = x In this way, the all out no. of balls = x+5 P(E) = (Number of good results/Total number of...

### A pass on is numbered so that its faces show the numbers 1, 2, 2, 3, 3, 6. It is tossed multiple times and the complete score in two tosses is noted. Complete the accompanying table which gives a couple of upsides of the absolute score on the two tosses:

What is the likelihood that the absolute score is (I) even? (ii) 6? (iii) something like 6? Solution: The table will be as per the following: 1 2 2 3 3 6 1 2 3 3 4 4 7 2 3 4 4 5 5 8 2 3 4 4 5 5 8 3...

### Two clients Shyam and Ekta are visiting a specific shop around the same time (Tuesday to Saturday). Each is similarly prone to visit the shop on any day as on one more day. What is the likelihood that both will visit the shop on

(I) that very day? (ii) back to back days? (iii) distinctive days? Solution: Since there are 5 days and both can go to the shop in 5 different ways each thus, The complete number of potential...

### A stash contains hundred 50p coins, fifty ₹1 coins, twenty ₹2 coins and ten ₹5 coins. In case almost certainly, one of the coins will drop out when the bank is flipped around, what is the likelihood that the coin

(I) will be a 50 p coin? (ii) won't be a ₹5 coin? Solution: Complete no. of coins = 100+50+20+10 = 180 P(E) = (Number of ideal results/Total number of results) (I) Total number of 50 p coin = 100 P...

### Gopi purchases a fish from a shop for his aquarium. The retailer takes out one fish indiscriminately from a tank containing 5 male fish and 8 female fish (see Fig. 15.4). What is the likelihood that the fish taken out is a male fish?

Solution: The all out number of fish in the tank = 5+8 = 13 All out number of male fish = 5 P(E) = (Number of good results/Total number of results) P (male fish) = 5/13 = 0.38

### A shot in the dark comprises of turning a bolt which stops pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 15.5), and these are similarly probable results. What is the likelihood that it will point at

(I) 8? (ii) an odd number? (iii) a number more noteworthy than 2? (iv) a number under 9? Solution: Complete number of potential results = 8 P(E) = (Number of great results/Total number of results)...

### The probability that a student is not a swimmer is . Then the probability that out of five students, four are swimmers is A. B. C. D. None of these

Solution:Answer: A. ${ }^{5} \mathrm{C}_{4} 1 / 5(4 / 5)^{4}$ Explanation: Assume that $X$ is the number of pupils who are swimmers out of a class of five. The Bernoulli trials are also used to...

### A bite the dust is tossed once. Discover the likelihood of getting

(I) an indivisible number; (ii) a number lying somewhere in the range of 2 and 6; (iii) an odd number. Solution: Complete potential occasions when a dice is tossed = 6 (1, 2, 3, 4, 5, and 6) P(E) =...

### In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is A. B. C. D.

Solution:Answer: C. $(9 / 10)^{5}$ Explanation: Assume $X$ is the number of times a random sample of 5 bulbs was selected for faulty bulbs. Bernoulli trials are also the process of repeatedly...

### One card is drawn from a very much rearranged deck of 52 cards. Discover the likelihood of getting

(I) a lord of red tone (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the sovereign of jewels Arrangement: All out number of potential results = 52 P(E) = (Number of...

### Five cards the ten, jack, sovereign, ruler and trick card, are very much rearranged with their face downwards. One card is then gotten up.

(I) What is the likelihood that the card is the sovereign? (ii) If the sovereign is drawn and set to the side, what is the likelihood that the subsequent card gotten is (a) an ace? (b) a sovereign?...

### 12 flawed pens are inadvertently blended in with 132 great ones. It is unimaginable to simply take a gander at a pen and tell whether it is blemished. One pen is taken out aimlessly from this parcel. Decide the likelihood that the pen taken out is a decent one.

Solution: Quantities of pens = Numbers of blemished pens + Numbers of good pens ∴ Total number of pens = 132+12 = 144 pens P(E) = (Number of great results/Total number of results) P(picking a decent...

### It is known that of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?

Solutions: Assume that $X$ is the number of times a defective article was selected from a random sample space of 12 articles. Bernoulli trials are also repeated articles in a random sample space....

### (I) A ton of 20 bulbs contain 4 blemished ones. One bulb is drawn indiscriminately from the part. What is the likelihood that this bulb is damaged?(ii) Suppose the bulb attracted (I) isn’t damaged and isn’t supplanted. Presently one bulb is drawn indiscriminately from the rest. What is the likelihood that this bulb isn’t deficient?

Solution: (I) Number of damaged bulbs = 4 The absolute number of bulbs = 20 P(E) = (Number of great results/Total number of results) ∴ Probability of getting a damaged bulb = P (deficient bulb) =...

### Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Solution: Let's say $X$ represents the amount of times you've gotten sixes in six die throws. Bernoulli trials are also the repeated tossing of die selection. As a result, the chances of getting six...

### Find the probability of getting 5 exactly twice in 7 throws of a die.

Solutions: Let's say $X$ represents the number of times you've gotten 5 in 7 die throws. The Bernoulli trials are also the repeated tossing of a die. Thus, probability of getting 5 in a single...

### A container contains 90 plates which are numbered from 1 to 90. On the off chance that one circle is drawn aimlessly from the crate, discover the likelihood that it bears

(I) a two-digit number (ii) an ideal square number (iii) a number distinguishable by 5. Solution: The complete number of plates = 90 P(E) = (Number of great results/Total number of results) (I)...

### A kid has a kick the bucket whose six faces show the letters as given underneath:

The bite the dust is tossed once. What is the likelihood of getting (I) A? (ii) D? Solution: The complete number of potential results (or occasions) = 6 P(E) = (Number of ideal results/Total number...

### A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is . What is the probability that he will win a prize (a) At least once (b) Exactly once (c) At least twice?

Solution: (a) Let $X$ be the number of prizes won in 50 lottery drawings, and the trials be Bernoulli trials.Here clearly, we have $\mathrm{X}$ is a binomial distribution where $\mathrm{n}=50$ and...

### Assume you drop a kick the bucket aimlessly on the rectangular locale displayed in Fig. 15.6. What is the likelihood that it will land inside the circle with breadth 1m?

Solution: To start with, compute the space of the square shape and the space of the circle. Here, the space of the square shape is the conceivable result and the space of the circle will be the...

### On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

Solution: Bernoulli trials are the repeated correct answer guessing from the given multiple choice questions in this question. Assume that $X$ represents the number of correct answers in the...

### Let and be independent events with and . Find (i) (ii) (iii) (iv)

Solution: Given: $\mathrm{P}(\mathrm{A})=0.3$ and $\mathrm{P}(\mathrm{B})=0.4$ (i) $P(A \cap B)$Given $P ( A )=0.3$ and $P ( B )=0.4$ When A and B are independent.Two events are independent,...

### Let and be independent events with and . Find (i) (ii) (iii) (iv)

Given $P ( A )=0.3$ and $P ( B )=0.4$ (i) $P(A \cap B)$ When A and B are independent. Two events are independent, statistically independent, or stochastically independent if the occurrence of one...

### Given that the events and are such that and . Find if they are (i) mutually exclusive (ii) independent.

Solution: Given: $\mathrm{P}(\mathrm{A})=1 / 2, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=1 / 5$ and let the probability of event be $\mathrm{P}(\mathrm{B})=\mathrm{p}$ (i) Mutually exclusive When A...

Solution: Given: A box of oranges. Let A, B, and C represent the events that occur when the first, second, and third drawn oranges are all excellent. Now, $P(A)=P$ (good orange in first draw) $=12 /... read more ### Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black. Given: A pack of 52 cards. As we all know, there are a total of 26 black cards. Let$A$and$B$represent the first and second drawn cards being black, respectively. Now if the probability of event... read more ### If and , find if and are independent events. Solution: Given:$P(A)=3 / 5$and$P(B)=1 / 5$As A and B are independent events.So we know that for independent events,$\Rightarrow P(A \cap B)=P(A) . P(B)$Substituiting the values we get,$=3 / 5...

Solution:Answer: D. $P(A)=P(B)$ Explanation: Given: $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{P}(\mathrm{B} \mid \mathrm{A})$We are aware that, according to the concept of conditional...