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Probability

Short Answer Type

1061. An unbiased coin is tossed six times. Find the probability of obtaining at least 2 heads.

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1062. In a box containing 100 bulbs, 10 are defective. What is the probabilities that out of a sample of 5 bulbs
(i) none is defective (ii) exactly two are defective.

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1063. If “getting a 6” in a throw of an unbiased die is a success and the random variable x denotes the number of successes in five throws of the die, find P (2 < x < 5)

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1064. If “getting a 5 or a 6” in the throw of an unbiased die is a success and the random variable x denotes the number of successes in 6 throws of the die, find P (x ≤ 4).

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Long Answer Type

1065. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the dice.

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Short Answer Type

1066. Eight coins are thrown simultaneously. Find the probability of getting at least six heads.

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Long Answer Type

1067.

A pair of dice is thrown 7 times. If getting a total of 7 is considered a success, what is the probability of
(i) no success? (ii) 6 successes? (iii) at least 6 successes? (iv) at most 6 successes?

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1068. Suppose X has a binomial distribution B $open parentheses 6 comma space 1 half close parentheses.$ Show that X = 3 is the most likely outcome.

Since $straight B space open parentheses 6 comma space 1 half close parentheses$ is binomial distribution
$therefore space space space space space straight n space equals space 6 comma space space space straight p space equals space 1 half comma space space straight q space equals space 1 space minus space straight p space equals space 1 minus space 1 half space equals space 1 half$
$straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight C presuperscript 6 subscript 0 space open parentheses 1 half close parentheses to the power of 6 space equals space 1 space cross times space 1 over 64 space equals 1 over 64 straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight C presuperscript 6 subscript 1 space open parentheses 1 half close parentheses to the power of 5 space open parentheses 1 half close parentheses space equals space 6 space cross times space 1 over 32 cross times space 1 half space equals space 6 over 64 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight C presuperscript 6 subscript 2 space open parentheses 1 half close parentheses to the power of 4 space open parentheses 1 half close parentheses squared space equals fraction numerator 6 space cross times space 5 over denominator 1 space cross times 2 end fraction cross times 1 over 16 cross times 1 fourth space equals 15 over 64 straight P left parenthesis straight X space equals 3 right parenthesis space equals space straight C presuperscript 6 subscript 3 space open parentheses 1 half close parentheses cubed space open parentheses 1 half close parentheses cubed space equals space fraction numerator 6 space cross times space 5 space cross times space 4 over denominator 1 space cross times space 2 space cross times 3 end fraction cross times space 1 over 8 cross times 1 over 8 space equals space 20 over 64 straight P left parenthesis straight X space equals space 4 right parenthesis space equals space straight C presuperscript 6 subscript 4 space open parentheses 1 half close parentheses squared space open parentheses 1 half close parentheses to the power of 4 space equals fraction numerator 6 space cross times 5 over denominator 1 cross times 2 end fraction cross times 1 fourth cross times 1 over 16 equals 15 over 64 straight P left parenthesis straight X space equals 5 right parenthesis space equals space straight C presuperscript 6 subscript 5 space open parentheses 1 half close parentheses space open parentheses 1 half close parentheses to the power of 5 space equals space 6 over 1 cross times 1 half cross times 1 over 32 equals space 6 over 64 straight P left parenthesis straight X space equals 6 right parenthesis space equals space straight C presuperscript 6 subscript 6 space open parentheses 1 half close parentheses to the power of 6 space equals space 1 space cross times space 1 over 64 space equals 1 over 64.$
Since $20 over 64$ is maximum of all the above values
$therefore space space space straight X space equals space 3 space is space the space most space likely space out space come. space$