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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.

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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.

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 thin space 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 space 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 space 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 comma 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

Mean X =

sum from blank to blank of straight X thin space straight P left parenthesis straight X right parenthesis

equals space fraction numerator 2 space cross times 1 plus 3 cross times 2 plus 4 cross times 3 plus 5 cross times 4 space plus space 6 space cross times 5 space plus space 7 space cross times 6 space plus space 8 space cross times 5 space plus space 9 cross times 4 space plus space 10 space cross times 3 space plus 11 space cross times 2 space plus 12 space cross times space 1 over denominator 36 end fraction space equals 252 over 36 space equals space 7

Variance

straight X space equals space sum from blank to blank of straight X squared space straight P left parenthesis straight X right parenthesis space minus space left parenthesis mean right parenthesis squared

fraction numerator left parenthesis 2 squared cross times 1 space plus space 3 squared space cross times 2 space plus space 4 squared cross times 3 space plus space 5 squared space cross times space 4 space plus space 6 squared space cross times space 5 space plus space 7 squared space cross times 6 space plus space 8 space squared space cross times 5 plus 9 squared space cross times 4 space plus space 10 squared space cross times 3 space plus space 11 squared cross times 2 space plus space 12 squared space cross times 1 right parenthesis over denominator 36 end fraction minus 7 squared

fraction numerator 4 plus 18 plus 48 plus 100 plus 180 plus 294 plus 320 plus 324 plus 300 plus 242 plus 144 over denominator 36 end fraction minus 49

1974 over 36 minus 49 space equals space fraction numerator 1974 minus 1764 over denominator 36 end fraction space equals 210 over 36 space equals space 35 over 6.

therefore space space straight S. straight D. space space equals space square root of Variance space equals square root of 35 over 6 end root space equals space 1.4

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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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