Binomial Probability Theorem

Q.

A coin is tossed $10$ times.

The probability of getting exactly six heads is

1. $\frac{512}{513}$

2. $\frac{105}{512}$

3. $\frac{100}{153}$

4. ${}^{10}C_6$

Q.

A sample of $4$ items is drawn at random without replacement from a lot of $10$ items containing $3$ defective. If $X$ denotes the number of defective items in the sample then $P\left(0<x<3\right)$ is equal to

1. $\frac{3}{10}$

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

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

4. $\frac{1}{6}$

Q. An experiment consists of three throws of coin and success means two heads. The probability of no success, if experiment is repeated three times, is

1. $\frac{1}{25}$
2. $\frac{2}{5}$
3. $\frac{24}{25}$
4. None

Q. Two players toss 4 coins each. The probability that they both obtain the same number of heads is

1. $\frac{5}{256}$
2. $\frac{1}{16}$
3. $\frac{35}{126}$
4. $\frac{3}{16}$

Q. A box has 50 pens of which 20 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement, at most one is defective?

1. ${\left(\frac{2}{5}\right)}^5$
2. $3.6\times {\left(\frac{3}{5}\right)}^4$
3. ${\left(\frac{3}{5}\right)}^5$
4. $2.6\times {\left(\frac{3}{5}\right)}^4$

Q. A fair coin is tossed a fixed number of times. If the probability of getting $7$ heads is equal to probability of getting $9$ heads, then the probability of getting $2$ heads is :

1. $\frac{15}{2^{12}}$
2. $\frac{15}{2^{13}}$
3. $\frac{15}{2^8}$
4. $\frac{15}{2^{14}}$

Q. Match the following by appropiately matching the lists based on the information given in Column I and Column II.
​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{(a) The probability of a bomb hitting a bridge is}& \\ \frac{1}{2}.\text{Two direct hits are needed to destroy it.}& \\ \text{The number of bombs recquired so that the}& \\ \text{probability of the bridge being destroyed is}& \\ \text{greater than}0.9\text{can be}& \text{(p)}4\\ \text{(b) A bag contains}2\text{red,}3\text{white,}5\text{black balls, a}& \\ \text{ball is drawn its color is noted and replaced}& \\ \text{The number of times, a ball can be drawn}& \\ \text{so that the probability of getting a red ball}& \\ \text{for the first time is atleast}1/2& \text{(q)}6\\ \text{(c) A drawer contains a mixture of red socks}& \\ \text{and blue socks, at most}17\text{in all. It so}& \\ \text{happens that when two socks are selected}& \\ \text{randomly without replacement, there is a}& \\ \text{probability of exactly}1/2\text{that both are red}& \\ \text{or both are blue. Then number of red}& \\ \text{socks in drawer can be}& \text{(r)}7\\ \text{(d) There are two red, two blue, two white and}& \\ \text{certain number (greater than}0\text{) of green}& \\ \text{socks in a drawer. If two socks are taken at}& \\ \text{random from the drawer without}& \\ \text{replacement, the probability that they}& \\ \text{are of same color is}\frac{1}{5}\text{, then the number of}& \\ \text{green socks are}& \text{(s)}10\end{array}$

1. $a\to r,s;b\to p,q,r,s;c\to p,q,r,s;d\to p$
2. $a\to r,s;b\to p,q,s;c\to p,q,r,s;d\to p$
3. $a\to r,s;b\to p,s;c\to p,s;d\to p$
4. $a\to r,s;b\to p,q,r,s;c\to p,q,r,s;d\to q,r,s$

Q. A coin whose faces marked 2 and 3 is thrown 5 times, then chance of obtaining a total of 12 is

1. $\frac{5}{16}$
2. $\frac{5}{8}$
3. $\frac{5}{32}$
4. $\frac{5}{24}$

Q. The probability of success is p in one trial and the experiment be repeated n times. If the ratio of the probability of exactly r success to the probability of exactly r failures is independent of n and r, then p =

1. $\frac{1}{2}$
2. exists no such value of p
3. $\frac{1}{3}$
4. None of these

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

Q.

In a game “odd man out” each of $m\ge 2$ person tosses a coin to determine who will buy refreshments for the entire group. The odd man out is the one with a different outcome from the rest. The probability that there is a loser in any game is.

1. $\frac{1}{2^{m-1}}$

2. $\frac{m-1}{2^{m-1}}$

3. $\frac{m}{2^{m-1}}$

4. none of the above

Q.

Numbers are selected at random, one at a time, form 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 the probability that the event E occurs at least 3 times.

1. $=\frac{87}{(25{)}^3}$

2. $=\frac{97}{(25{)}^5}$

3. $=\frac{87}{(25{)}^4}$

4. $=\frac{97}{(25{)}^4}$

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

Q. One hundred identical coins, each with probability p of showing up heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

1. $\frac{49}{101}$
2. $\frac{50}{101}$
3. $\frac{1}{2}$
4. $\frac{51}{101}$

Q. There are two die $A$ and $B$ both having six faces. Die $A$ has three faces marked with $1$, two faces marked with $2$, and one face marked with $3$. Die $B$ has one face marked with $1$, two faces marked with $2$, and three faces marked with $3$. Both dices are thrown randomly once. If $E$ be the event of getting sum of the numbers appearing on top faces equal to $x$, let $P\left(E\right)$ be the probability of event $E$, then
$P\left(E\right)$ is minimum when $x$ equals to

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

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 variance of the number obtained on a throw of an unbiased die is:

1. $\frac{35}{12}$
2. $\frac{12}{35}$
3. $\frac{7}{35}$
4. $\frac{9}{35}$

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

Q. A fair coin is tossed n times and let X denote the number of heads obtained. If P(X = 4), P(x=5) and (x=6) are in A.P., then n is equal to:

1. 6 and 7
2. 6 and 14
3. 7 or 14
4. None

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. 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. A biased coin is thrown 100 times with probability of head = p and probability of tail = q. The probability of getting 55 heads in 100 trials is $\frac{100!}{55!\times 45!}{p}^{55}{q}^{45}$

1. True
2. False