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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.

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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.

Let X denote the sum of the numbers obtained on the two dice. X can take values 2, 3, ....., 12

straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 1 right parenthesis right curly bracket space equals space 1 over 36 straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 2 right parenthesis comma space left parenthesis 2 comma space 1 right parenthesis right curly bracket space equals space 2 over 36 straight P left parenthesis straight X space equals space 4 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 3 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 3 comma space 1 right parenthesis right curly bracket space equals 3 over 36 straight P left parenthesis straight X space equals space 5 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 4 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis comma space left parenthesis 4 comma space 1 right parenthesis right curly bracket space equals 4 over 36 straight P left parenthesis straight X space equals space 6 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 5 right parenthesis comma space left parenthesis 2 comma space 4 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis comma space left parenthesis 4 comma space 2 right parenthesis comma space left parenthesis 5 comma space 1 right parenthesis right curly bracket space equals space 5 over 36 straight P left parenthesis straight X space equals space 7 right parenthesis space equals space straight P left curly bracket left parenthesis 1 comma space 6 right parenthesis comma space left parenthesis 2 comma space 5 right parenthesis comma space left parenthesis 3 comma space 4 right parenthesis comma space left parenthesis 4 comma space 3 right parenthesis comma space left parenthesis 5 comma space 2 right parenthesis comma space left parenthesis 6 comma space 1 right parenthesis right curly bracket space equals space 6 over 36 straight P left parenthesis straight X space equals space 8 right parenthesis space equals space straight P left curly bracket left parenthesis 2 comma space 6 right parenthesis comma space left parenthesis 3 comma space 5 right parenthesis comma space left parenthesis 4 comma space 4 right parenthesis comma space left parenthesis 5 comma space 3 right parenthesis comma space left parenthesis 6 comma space 2 right parenthesis right curly bracket space equals space 5 over 36 straight P left parenthesis straight X space equals space 9 right parenthesis space equals space straight P left curly bracket left parenthesis 3 comma space 6 right parenthesis comma space left parenthesis 4 comma space 5 right parenthesis comma space left parenthesis 5 comma space 4 right parenthesis comma space left parenthesis 6 comma space 3 right parenthesis right curly bracket space equals space 4 over 36 straight P left parenthesis straight X space equals space 10 right parenthesis space equals space straight P left curly bracket left parenthesis 4 comma space 6 right parenthesis space left parenthesis 5 comma space 5 right parenthesis comma space left parenthesis 6 comma space 4 right parenthesis right curly bracket space equals space 3 over 36 straight P left parenthesis straight X space equals space 11 right parenthesis space equals space straight P left curly bracket left parenthesis 5 comma space 6 right parenthesis comma space left parenthesis 6 comma space 5 right parenthesis right curly bracket space equals space 2 over 36 straight P left parenthesis straight X space equals space 12 right parenthesis space equals space straight P left curly bracket left parenthesis 6 comma space 6 right parenthesis right curly bracket space equals space 1 over 36

Probability distribution of X is

therefore space space straight mu space equals space straight E left parenthesis straight X right parenthesis space equals space sum from straight i space equals space 1 to straight n of space straight x subscript straight i space straight p subscript straight i space equals space 2 space cross times 1 over 36 plus 3 cross times 2 over 36 plus 4 cross times 3 over 36 plus 5 cross times 4 over 36 space space space space space space space space space space space plus space 6 space cross times 5 over 36 plus 7 cross times 6 over 36 plus 8 cross times 5 over 36 plus 9 cross times 4 over 36 plus 10 cross times 3 over 36 plus 11 cross times 2 over 36 plus 12 space cross times 1 over 36 space equals fraction numerator 2 plus 6 plus 12 plus 20 plus 30 plus 42 plus 40 plus 36 plus 30 plus 22 plus 12 over denominator 36 end fraction space equals 252 over 36 space equals 7

∴ the mean of the sum of the numbers that appear on throwing two fair dice is 7.

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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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