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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.

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

Here X denotes the number of heads obtained in three tosses of a coin. Thus X can take the values 0, 1, 2, 3.
Let p be the probabilities of getting a head and q be the probability of not getting a head,
$therefore space space space space straight p space equals space 1 half comma space space straight q space equals space 1 space minus space straight p space equals space 1 space space minus 1 half space equals space 1 half$
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight q space cross times space straight q space cross times space straight q space space equals space 1 half cross times 1 half cross times 1 half space equals 1 over 8 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight p space straight q space straight q space space plus space straight q space straight p space straight q space plus space straight q space space straight q space straight p space space space space space space space space space space space space space space space equals space 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half space equals 1 over 8 plus 1 over 8 plus 1 over 8 space equals 3 over 8 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space straight p space straight q space plus space straight p space straight q space straight p space plus space straight q space straight p space straight p space space space space space space space space space space space space space space space equals space 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half plus 1 half cross times 1 half cross times 1 half space equals 1 over 8 plus 1 over 8 plus 1 over 8 space equals 3 over 8 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space straight p space straight p space equals space 1 half cross times 1 half cross times 1 half space equals 1 over 8$
∴ probability distribution table is

$therefore space space space straight mu space equals space sum from blank to blank of space straight x space straight p subscript straight i space equals space 3 over 8 plus 6 over 8 plus 3 over 8 space equals 12 over 8 space equals 3 over 2 space equals space 1.5$

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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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