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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.

Let X denote the number of kings obtained in two draws. X takes the values 0, 1, 2.
Let p denote probability of first king.

therefore space space space straight p space equals space 4 over 52 comma space space space space straight q space equals space 48 over 52

straight P left parenthesis straight X space equals space 0 right parenthesis space equals space 48 over 52 cross times 47 over 51 space equals 12 over 13 cross times 47 over 51 space equals space 188 over 221 straight P left parenthesis straight X space equals 1 right parenthesis space equals space 4 over 52 cross times 48 over 51 plus 48 over 52 cross times 4 over 51 space equals 16 over 221 plus 16 over 221 equals space 32 over 221 straight P left parenthesis straight X space equals 2 right parenthesis space equals space 4 over 52 cross times 3 over 51 space equals 1 over 31 cross times 1 over 17 space equals 1 over 221

∴ probability distribution is

Now, Mean of X = E(X) =

sum from straight i space equals space 1 to straight n of straight x subscript straight i space straight p left parenthesis straight x subscript straight i right parenthesis

equals space 0 cross times 188 over 121 plus 1 cross times 32 over 221 plus 2 cross times 1 over 221 space equals space 0 plus 32 over 221 plus 2 over 221 space equals 34 over 221

Also,

straight E left parenthesis straight X squared right parenthesis space equals space sum from straight i space equals space 1 to straight n of space straight x subscript straight i squared space straight p left parenthesis straight x subscript straight i right parenthesis

equals space 0 squared cross times 188 over 121 plus 1 squared cross times 32 over 221 plus 2 squared cross times 1 over 221 equals space 0 space plus space 32 over 221 plus 4 over 221 space equals 36 over 221

therefore space space space space space Var space left parenthesis straight X right parenthesis space equals space straight E left parenthesis straight X right parenthesis squared space minus space left square bracket straight E left parenthesis straight X right parenthesis right square bracket squared space space space space space space space space space space space space space space space space space space space space space space space equals space 36 over 221 space minus space open parentheses 34 over 221 close parentheses squared equals space 36 over 221 space minus space 1156 over 48841 space equals space fraction numerator 7956 minus 1156 over denominator 48841 end fraction space equals 6800 over 48841

therefore space space space space space straight sigma squared space equals space square root of Var left parenthesis straight X right parenthesis end root space equals space fraction numerator square root of 6800 over denominator 221 end fraction space equals space fraction numerator 82.46 over denominator 221 end fraction space equals space 0.37

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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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