Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let X denote the occurrence of 3,4 or 5 in a single die. Then, X follows binomial distribution with n=5.
Let p=probability of getting 3,4, or 5 in a single die .
p = \[\frac{3}{6} = \frac{1}{2}\]
\[q = 1 - \frac{1}{2} = \frac{1}{2} \]
\[P(X = r) = ^{5}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{5 - r} \]
\[P(\text{ at least 3 successes } ) = P(X \geq 3) \]
\[ = P(X = 3) + P(X = 4) + P(X = 5)\]
\[ = ^{5}{}{C}_3 \left( \frac{1}{2} \right)^3 \left( \frac{1}{2} \right)^{5 - 3} + ^{5}{}{C}_4 \left( \frac{1}{2} \right)^4 \left( \frac{1}{2} \right)^{5 - 4} +^{5}{}{C}_5 \left( \frac{1}{2} \right)^5 \left( \frac{1}{2} \right)^{5 - 5} \]
\[ = \frac{^{5}{}{C}_3 + ^{5}{}{C}_4 + ^{5}{}{C}_5}{2^5}\]
\[ = \frac{1}{2}\]
APPEARS IN
संबंधित प्रश्न
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
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs none will fuse after 150 days of use
An urn contains four white and three red balls. Find the probability distribution of the number of red balls in three draws with replacement from the urn.
The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90% ?
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 .
A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
Find the binomial distribution whose mean is 5 and variance \[\frac{10}{3} .\]
If on an average 9 ships out of 10 arrive safely at ports, find the mean and S.D. of the ships returning safely out of a total of 500 ships.
The mean and variance of a binomial variate with parameters n and p are 16 and 8, respectively. Find P (X = 0), P (X = 1) and P (X ≥ 2).
Find the expected number of boys in a family with 8 children, assuming the sex distribution to be equally probable.
A dice is thrown thrice. A success is 1 or 6 in a throw. Find the mean and variance of the number of successes.
If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P (X ≤ 5).
If X follows a binomial distribution with mean 4 and variance 2, find P (X ≥ 5).
If the mean of a binomial distribution is 20 and its standard deviation is 4, find p.
The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.
If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire in order to have more than 50% chance of hitting it at least once is
Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X= 5) and P (X = 6) are in AP, the value of n 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 five-digit number is written down at random. The probability that the number is divisible by 5, and no two consecutive digits are identical, is
Mark the correct alternative in the following question:
Which one is not a requirement of a binomial dstribution?
Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that none is a spade ?
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs not more than one will fuse after 150 days of use
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs at least one will fuse after 150 days of use
Determine the binomial distribution where mean is 9 and standard deviation is `3/2` Also, find the probability of obtaining at most one success.
Find the mean and variance of the random variable X which denotes the number of doublets in four throws of a pair of dice.
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
A pair of dice is thrown four times. If getting a doublet is considered a success then find the probability of two success.
The mean and variance of a binomial distribution are α and `α/3` respectively. If P(X = 1) = `4/243`, then P(X = 4 or 5) is equal to ______.
If a random variable X follows the Binomial distribution B(5, p) such that P(X = 0) = P(X = 1), then `(P(X = 2))/(P(X = 3))` is equal to ______.
An experiment succeeds thrice as often as it fails. Then in next five trials, find the probability that there will be two successes.
