Probability Questions
Q1. Ten fair coins are tossed simultaneously. Find the probability of getting atleast seven heads. Q2. You have two jars, 50 red an 50 blue marbles. You can distribute the marbles in the two jars as you wish. Then one of the two jars is chosen at random, and finally a marble is picked at random from the chosen jar. How do you maximize the probability that a red marble is picked? Q3. The probability of rain was 40%. If it rained, the Redskins had a 30% chance of winning, if it did not rain, they had a 55% chance of winning. Given that the Redskins won, what is the probability that it rained? Q4. Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelope is Q5. Two athletes A and B participate in a race along with other athletes. If the chance of A wining the race is 1/6 and that of B wining the same race is 1/8, then the chance that neither wins the race is Q6. A is one of the six race horses which is to be ridden by one of the two jockeys B or C. It is 2 :1 that B rides A in which case all the horses are equally likely to win but if C rides, then A’s chances of wining are trebled. The odds against his winning are Q7.A and B play by throwing a pair of dice alternately. A wins if he throws 6 before B throws 7. If A starts the game their chances of wining the game are in the ratio
(a) 1/6 (b) 5/6 (c) 1/3 (d) 2/3
(a) 13 : (b) 5 : 13 (c) 8 : 5 (d) 5 : 8
(a) 28 : 33 (b) 29 : 32 (c) 30 : 31 (d) none
Q3. The probability of rain was 40%. If it rained, the Redskins had a 30% chance of winning, if it did not rain, they had a 55% chance of winning. Given that the Redskins won, what is the probability that it rained?
Sol.
{(4/10)(3/10)}/ {{(4/10)(3/10) + (6/10)(55/100)}
= 4/15
Q4. Three letters, to each of which corresponds an envelope, are placed in the envelopes at random. The probability that all the letters are not placed in the right envelope is
(a) 1/6 (b) 5/6 (c) 1/3 (d) 2/3
Sol.
Ways of having all 3 worng out 3 letters to 3 envelops
= 3! - {When all three are correct + when one is correct and two are worng} = 2
probability = 1/3
Q5. Two athletes A and B participate in a race along with other athletes. If the chance of A wining the race is 1/6 and that of B wining the same race is 1/8, then the chance that neither wins the race is
Sol.
= 1 - {(1/6)(7/8) + (5/6)(1/8) - (1/6)(1/8)}
= 37/48
Q7.A and B play by throwing a pair of dice alternately. A wins if he throws 6 before B throws 7. If A starts the game their chances of wining the game are in the ratio
(a) 28 : 33 (b) 29 : 32 (c) 30 : 31 (d) none
Sol.
Probability of wining of A = P(A) + P(A')P(B')P(A) + P(A')P(B')P(A')P(B')P(A) +..........
= (5/36) + (31/36)(30/36)(5/36) + (31/36)(30/36)(31/36)(30/36)(5/36)+............
= (5/36){1 + (31x30/36x36)2 + (31x30/36x36)4+....................}
= 30/61
Probability of wining of B = 1 - (30/61) = 31/61
ratio = 30:31
Q2. You have two jars, 50 red an 50 blue marbles. You can distribute the marbles in the two jars as you wish. Then one of the two jars is chosen at random, and finally a marble is picked at random from the chosen jar. How do you maximize the probability that a red marble is picked?
Sol.
(1/2)(49/99) +(1/2)(1) = 74/99
Thanks for solving and posting the answers of all the questions.
I have been searching for detailed solutions to some questions in "The Pearson guide to QUANT for CAT by Nishit Sinha".If any of you have the detaied answers,please help me out.
Probability
1.A three digit number is to be formed by using the digits 1,2....9.What is the probability that the number formed is greater than 500, if repetition is not allowed?
2.The probability of a number being solved by A is 1/4 and B solving it is 1/5 the.If they try independently what is the probability that the problem is not solved?
3.Ten horses are running in a race,chances that A will win is 30%,that B will win are 20% and C will win are 10%.What is the probabilty that one of them will win?
4.A's chance of winning a single game against B is 3/4.Find the chance that in a series of 5 games with B,A wins exactly 3 games?
5.It is known that at noon,the sun is hidden by clouds on an average of 2 days out of every 3.Find the probability that at noon on atleast 4 out of 5 days,the sun will be shining.
6.Nazneen was dialing a telephone number of 6 digits.She forgot the last 3 digits but rememberd that te number formed by the last 3 digits in the same order was a perfect square.What is the probability that she dialed a right number?
7.What is the probability that there are 53 sundays and 53 tuesdays in a leap year?
8.What is the probability that there are 53 sundays and 53 saturdays in a leap year?
Q1. Ten fair coins are tossed simultaneously. Find the probability of getting atleast seven heads.
Sol.
(10C7 + 10C8 + 10C9 + 10C10)/ 210 = 11/64