Advertisements
Advertisements
Question
The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?
Solution
Let X be number of times the target is hit. Then, X follows a binomial distribution with n =7, \[p = \frac{1}{4}\text{ and } q = \frac{3}{4}\]
\[P(X = r) = ^{7}{}{C}_r \left( \frac{1}{4} \right)^r \left( \frac{3}{4} \right)^{7 - r} \]
\[P( \text{ hitting the target at least twice} )\]
\[ = P(X \geq 2) \]
\[ = 1 - \left\{ P(X = 0) + P(X = 1) \right\}\]
\[ = 1 -^{7}{}{C}_0 \left( \frac{1}{4} \right)^0 \left( \frac{3}{4} \right)^{7 - 0} - ^{7}{}{C}_1 \left( \frac{1}{4} \right)^1 \left( \frac{3}{4} \right)^{7 - 1} \]
\[ = 1 - \left( \frac{3}{4} \right)^7 - 7\left( \frac{1}{4} \right) \left( \frac{3}{4} \right)^6 \]
\[ = 1 - \frac{1}{16384}(2187 + 5103) \]
\[ = 1 - \frac{3645}{8192}\]
\[ = \frac{4547}{8192}\]
APPEARS IN
RELATED QUESTIONS
Given that X ~ B(n= 10, p). If E(X) = 8 then the value of
p is ...........
(a) 0.6
(b) 0.7
(c) 0.8
(d) 0.4
Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).
A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.
If getting 5 or 6 in a throw of an unbiased die is a success and the random variable X denotes the number of successes in six throws of the die, find P (X ≥ 4).
A bag contains 7 green, 4 white and 5 red balls. If four balls are drawn one by one with replacement, what is the probability that one is red?
Find the probability distribution of the number of sixes in three tosses of a die.
Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least 3 successes.
The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is
\[\frac{28 \times 9^6}{{10}^8} .\]
A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that (i) there is at least an even chance of drawing a heart (ii) the probability of drawing a heart is greater than 3/4?
Six coins are tossed simultaneously. Find the probability of getting
(i) 3 heads
(ii) no heads
(iii) at least one head
The probability that a certain kind of component will survive a given shock test is \[\frac{3}{4} .\] Find the probability that among 5 components tested exactly 2 will survive .
Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that exactly 2 will strike the target .
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that more than 3 contract the disease .
The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university all will graduate .
Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.
The probability of a shooter hitting a target is \[\frac{3}{4} .\] How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?
How many times must a man toss a fair coin so that the probability of having at least one head is more than 80% ?
A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.
A factory produces bulbs. The probability that one bulb is defective is \[\frac{1}{50}\] and they are packed in boxes of 10. From a single box, find the probability that none of the bulbs is defective .
Can the mean of a binomial distribution be less than its variance?
Find the binomial distribution whose mean is 5 and variance \[\frac{10}{3} .\]
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
A die is thrown three times. Let X be 'the number of twos seen'. Find the expectation of X.
If in a binomial distribution n = 4 and P (X = 0) = \[\frac{16}{81}\] , find q.
If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.
If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.
A fair coin is tossed 100 times. The probability of getting tails an odd number of times is
A biased coin with probability p, 0 < p < 1, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then p equals
A fair die is tossed eight times. The probability that a third six is observed in the eighth throw is
Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is
A coin is tossed 4 times. The probability that at least one head turns up is
The probability of selecting a male or a female is same. If the probability that in an office of n persons (n − 1) males being selected is \[\frac{3}{2^{10}}\] , the value of n is
For X ~ B(n, p) and P(X = x) = `""^8"C"_x(1/2)^x (1/2)^(8 - x)`, then state value of n and p
Which one is not a requirement of a binomial distribution?
If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9, P(X = 3), then p = ______.
If in the binomial expansion of (1 + x)n where n is a natural number, the coefficients of the 5th, 6th and 7th terms are in A.P., then n is equal to:
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:-
A pair of dice is thrown four times. If getting a doublet is considered a success then find the probability of two success.
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is ______.
