A coin is tossed 100 times. What is the proba | vedclass

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(C) Let $n = 100$ be the number of trials,$p = \frac{1}{2}$ be the probability of getting a head,and $q = \frac{1}{2}$ be the probability of getting a

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$\frac{1}{2}$

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(C) Let $n = 100$ be the number of trials,$p = \frac{1}{2}$ be the probability of getting a head,and $q = \frac{1}{2}$ be the probability of getting a tail. The probability of getting a head an odd number of times is given by the sum of probabilities for $r = 1, 3, 5, \dots, 99$. $P(r 	ext{ is odd}) = \sum_{r \in \{1, 3, \dots, 99\}} \binom{100}{r} p^r q^{100-r}$ $P(r 	ext{ is odd}) = \sum_{r \in \{1, 3, \dots, 99\}} \binom{100}{r} \left(\frac{1}{2}ight)^r \left(\frac{1}{2}ight)^{100-r}$ $P(r 	ext{ is odd}) = \left(\frac{1}{2}ight)^{100} \sum_{r \in \{1, 3, \dots, 99\}} \binom{100}{r}$ We know that the sum of binomial coefficients for odd $r$ is $2^{n-1}$. Thus,$\sum_{r \in \{1, 3, \dots, 99\}} \binom{100}{r} = 2^{100-1} = 2^{99}$. Therefore,$P(r 	ext{ is odd}) = \left(\frac{1}{2}ight)^{100} 	imes 2^{99} = \frac{2^{99}}{2^{100}} = \frac{1}{2}$.

$A$ coin is tossed $100$ times. What is the probability of getting a head an odd number of times?

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