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A coin is tossed 13 times.a)   How many different outcomes are possible?b)  How many different outcomes have exactly 9 heads?c)   How many different outcomes have at least 2 heads ?d)   How many different outcomes have at most 9 heads?

Question

A coin is tossed 13 times.a)   How many different outcomes are possible?b)  How many different outcomes have exactly 9 heads?c)   How many different outcomes have at least 2 heads ?d)   How many different outcomes have at most 9 heads?

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Solution

a) The number of different outcomes when a coin is tossed n times is given by 2^n. So, when a coin is tossed 13 times, the number of different outcomes is 2^13 = 8192.

b) The number of different outcomes with exactly 9 heads when a coin is tossed 13 times is given by the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of trials, k is the number of successful trials, and '!' denotes factorial. So, C(13, 9) = 13! / [9!(13-9)!] = 715.

c) The number of different outcomes with at least 2 heads when a coin is tossed 13 times is the total number of outcomes minus the outcomes with 0 or 1 head. So, it's 2^13 - C(13, 0) - C(13, 1) = 8192 - 1 - 13 = 8178.

d) The number of different outcomes with at most 9 heads when a coin is tossed 13 times is the sum of outcomes with 0 to 9 heads. So, it's C(13, 0) + C(13, 1) + C(13, 2) + ... + C(13, 9) = 1 + 13 + 78 + 286 + 715 + 1287 + 1716 + 1716 + 1287 + 715 = 7814.

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