Binomial Probability Theorem

Q.

A die is thrown $100$ times, and getting an even number is considered a success.

The variance of the number of successes is

1. $10$

2. $25$

3. $18$

4. $11$

Q. An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. [CBSE 2015]

Q.

The items produced by a firm are supposed to contain $5\%$ defective items. The probability that a sample of $8$ items will contain less than $2$ defective item is

1. $\left(\frac{27}{20}\right){\left(\frac{19}{20}\right)}^7$

2. $\left(\frac{533}{400}\right){\left(\frac{19}{20}\right)}^6$

3. $\left(\frac{153}{20}\right){\left(\frac{1}{20}\right)}^7$

4. $\left(\frac{35}{16}\right){\left(\frac{1}{20}\right)}^6$

Q. If 12 identical balls are to be placed in 3 different boxes, then the probability that a particular box contains exactly 3 balls, is?

1. $\frac{55}{3}(\frac{2}{3}{)}^{11}$
2. $55(\frac{2}{3}{)}^{10}$
3. $220(\frac{1}{3}{)}^{12}$
4. $22(\frac{1}{3}{)}^{11}$

Q.

There are n letters and n addressed envelopes. The probability that all the letters are not kept in the right envelope, is

1. 

2. None of these

3. 

4. 

Q. The probability of India winning a test match against West Indies is $\frac{1}{2}$. Assuming independence from match to match, the probability that in a five match series India’s second win occurs at third test is

1. 
2. 
3. 
4. 

Q.

If the mean and variance of a random variable $X$ having a Binomial distribution are $4$ and $2$, respectively, then find the value of $P(X=1)$

1. $\frac{1}{4}$

2. $\frac{1}{16}$

3. $\frac{1}{8}$

4. $\frac{1}{32}$

Q.

Out of $11$consecutives natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common differences, is:

1. $\frac{10}{99}$

2. $\frac{5}{33}$

3. $\frac{15}{101}$

4. $\frac{5}{101}$

Q.

A die is thrown 6 times. If getting an odd number is a success, What is the probability of

(i) sucessses?

(ii) atleast 5 sucesses?

(iii) atmost 5 sucesses?

Q.

If the mean and variance of a Binomial variable $X$ are $2$ and $1$, respectively, the $P(X\ge 1)$ is equal to

1. $\frac{2}{3}$

2. $\frac{4}{5}$

3. $\frac{7}{8}$

4. $\frac{15}{16}$

Q. A boy is throwing stones at a target. The probability of hitting the target at any trial is $\frac{1}{2}$ .The probability of hitting the target $5^{th}$ time at the ${10}^{th}$ throw is

1. $\frac{5}{2^{10}}$
2. $\frac{63}{2^9}$
3. $\frac{{}^{10}{C}_5}{2^{10}}$
4. $\frac{{}^{10}{C}_4}{2^{10}}$

Q.

The probability of a bomb hitting a bridge is $\frac{1}{2}$ and two direct hits are needed to destroy it. The least number of bombs required so that the probability of the bridge being destroyed is greater than $0.9$ is

1. $8$

2. $7$

3. $6$

4. $9$

Q. Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99\%$ is :

1. $5$
2. $6$
3. $7$
4. $8$

Q. Two players A and B play a game which consist of a series of matches independent of each other. Whoever first wins two matches, not necessarily consecutive, will win the game. The probability of A's winning, drawing or losing a match against B are $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ respectively. Which of the following is/are true?

1. Probability of A winning the game at the end of ${11}^{\text{th}}$ match is ${}^{10}{C}_1{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{3}\right)}^9+{}^{10}{C}_2{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{3}\right)}^9$
2. Probability of A winning the game at the end of ${11}^{\text{th}}$ match is ${}^{10}{C}_1{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{3}\right)}^9+{}^{10}{C}_2\left(\frac{1}{2}\right)\left(\frac{1}{6}\right){\left(\frac{1}{3}\right)}^9$
3. If it is known that A wins the game at the end of ${11}^{\text{th}}$ match, then the probability that B will win only one match is $\frac{9}{11}$
4. If it is known that A win the game at the end of ${11}^{th}$ match then the probability that B will win only one match is $\frac{9}{13}$

Q.

A car is parked by an owner amongst $25$ cars in a row, not at either end. On the return, he finds that exactly $15$ places are still occupied. The probability that both the neighboring places are empty.

1. $\frac{91}{276}$

2. $\frac{15}{184}$

3. $\frac{15}{92}$

4. None of these

Q.

A fair coin is tossed $100$ times the probability of getting tails an odd number of time

1. $\frac{1}{2}$

2. $\frac{1}{4}$

3. $0$

4. $1$

Q.

How many times must a man toss a fair coin so that the probability of having atleast one head is more than $90\%$?

Q. Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02, ..., 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

1. $\frac{1}{{25}^4}$
2. $\frac{96}{{25}^4}$
3. $\frac{97}{{25}^4}$
4. $\frac{95}{{25}^4}$

Q. Let the random variable $X$ have a binomial distribution with mean $8$ and variance $4.$ If $P(X\le 2)=\frac{k}{2^{16}},$ then $k$ is equal to:

1. $1$
2. $17$
3. $121$
4. $137$

Q.

A coin is tossed $n$ times. The probability of getting head atleast once is greater than $0.8$, then the least value of $n$ is

1. $2$

2. $3$

3. $5$

4. $4$

Q. The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chances of the event are

1. 
2. 
3. 
   
4. 

Q.

Two balls are drawn at random from a bag containing 2 white, 3 red. 5 green and 4 black balls, one by one without, replacement. Find the probabilitythat both the balls are of different colours.

Q. For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problems is :

1. $\frac{316}{25}{\left(\frac{4}{5}\right)}^{48}$
2. $\frac{164}{25}{\left(\frac{1}{5}\right)}^{48}$
3. $\frac{54}{5}{\left(\frac{4}{5}\right)}^{49}$
4. $\frac{201}{5}{\left(\frac{1}{5}\right)}^{49}$

Q. The probability that $A$ hits a target is $\frac{3}{4}$. Then the minimum number of attempts $A$ should take so that the probability of hitting the target at least once is at least $0.99$, is

Q.

In a game, a man wins a rupee for a six and losses a rupee for any other number when a fair die is thrown. The man decide to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.

Q. Let $X$ be a binomially distributed random variable with mean $4$ and variance $\frac{4}{3}$. Then, $54P(X\le 2)$ is equal to

1. $\frac{146}{27}$
2. $\frac{126}{81}$
3. $\frac{146}{81}$
4. $\frac{73}{27}$

Q. A problem in mathematics is given to $4$ students whose chances of solving individually are $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and $\frac{1}{5}$. The probability that the problem will be solved at least by one student is

1. $\frac{2}{3}$
2. $\frac{3}{5}$
3. $\frac{4}{5}$
4. $\frac{3}{4}$

Q.

In a bombing attack, there is $50\%$ chance that a bomb will hit the target.

At least two independent hits are required to destroy the target completely.

Then the minimum number of bombs, that must be dropped to ensure that there is at least $99\%$ chance of completely destroying the target, is

Q.

Team A plays with $5$ other teams exactly once. Assuming that for each match the probabilities of a win, draw and loss are equal, then

1. The probability that A wins and loses equal number of matches is $\frac{34}{81}$
2. The probability that A wins more number of matches than it loses is $\frac{17}{81}$
3. The probability that A wins and loses equal number of matches is $\frac{17}{81}$
4. The probability that A loses more number of matches than it wins is $\frac{16}{81}$

Q. The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least $90$% is :

1. $2$
2. $3$
3. $4$
4. $5$