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Probability

Short Answer Type

971. Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as:
six appears on at least one die

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972. Find the mean μ, variance σ² for the following probability distribution:

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973. Find the mean μ, variance σ² for the following probability distribution:

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Long Answer Type

974. Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

When a dice is rolled once, probability of obtaining six = $1 over 6$
and probability of obtaining a non-six = $5 over 6$
Now X can take values 0, 1, 2.
P(X = 0) = P(six does not appear a die) = P(non-six on both dice)
$equals space 5 over 6 cross times 5 over 6 space equals space 25 over 36$
P(X = 1) = P(six appear exactly on one die)
= P(six on first die and non-six on the second die)
+ P(six on the second die and non-six on the first die)
$equals space 1 over 6 cross times 5 over 6 plus 1 over 6 cross times 5 over 6 space equals space 5 over 36 plus 5 over 36 space equals 10 over 36$

P(X = 2) = P(six on both the dice) = $1 over 6 cross times 1 over 6 space equals 1 over 36$
∴ probability distribution of X is

∴ Expectation of X = mean of the variable X
$equals space sum from blank to blank of XP left parenthesis straight X right parenthesis space equals space 0 space cross times 25 over 36 plus 1 cross times 10 over 36 plus 2 cross times 1 over 36 equals space 0 plus 10 over 36 plus 2 over 36 space equals 12 over 36 space equals 1 third.$

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975. Find the mean number of heads in three tosses of a fair coin.

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976. Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).

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977. Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X.

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978. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings.

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979. Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.

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980. A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19, 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X ? Find mean, variance and standard deviation of X.

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