Bernoulli trials and the binomial distribution — practice questions

Question 1 · difficulty L2 · understanding

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

• A. $\frac{73}{10^8}$
• B. $\frac{67}{10^8}$
• C. $\frac{7}{10^7}$
• D. $\frac{81}{10^8}$

Question 2 · difficulty L2 · understanding

If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then $${{P(X = 2)} \over {P(X = 3)}}$$ is equal to :

• A. 1
• B. 10
• C. 25
• D. 5

Question 3 · difficulty L2 · understanding

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

• A. $${1 \over 8}$$
• B. $${5 \over 8}$$
• C. $${5 \over 16}$$
• D. 1

Question 4 · difficulty L3 · understanding

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96 p$ is equal to $\_\_\_\_$ .

Question 5 · difficulty L3 · understanding

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $$\frac{1}{3}$$ and $$\frac{2}{3}$$ respectively. Let $$x$$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $$\mathrm{P}(|x-y| \leq 2)$$ is $$p$$, then $$3^9 p$$ equals _________.

Question 6 · difficulty L3 · understanding

The sum and product of the mean and variance of a binomial distribution are 82.5 and 1350 respectively. Then the number of trials in the binomial distribution is ____________.

Question 7 · difficulty L3 · understanding

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $${3 \over 4}$$ and the remaining 6 questions correctly with probability $${1 \over 4}$$. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is $${{{{27}k}} \over {{4^{10}}}}$$, then k is equal to ___________.

Question 8 · difficulty L3 · understanding

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is ............

Question 9 · difficulty L3 · understanding

The random variable $$\mathrm{X}$$ follows binomial distribution $$\mathrm{B}(\mathrm{n}, \mathrm{p})$$, for which the difference of the mean and the variance is 1 . If $$2 \mathrm{P}(\mathrm{X}=2)=3 \mathrm{P}(\mathrm{X}=1)$$, then $$n^{2} \mathrm{P}(\mathrm{X}>1)$$ is equal to :

• A. 15
• B. 12
• C. 11
• D. 16

Question 10 · difficulty L3 · understanding

Let a die be rolled $$n$$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $$\frac{k}{2^{15}}$$, then $$\mathrm{k}$$ is equal to :

• A. 15
• B. 60
• C. 30
• D. 90