Answer
(a) $0.0009765625$
(b) $0.0009765625$
(c) $0.24609375$
Work Step by Step
(a) Let $p$ be the probability that a single bit is 1, which is given as $p = 0.5$. We can model the number of bits that are 1 as a binomial random variable $X$ with parameters $n = 10$ and $p = 0.5$.
The probability that all bits are 1s is given by $P(X = 10)$. Using the binomial probability formula, we have:
$P (X=10)=(\frac{10}{10})(0.5)^{10}(1-0.5)^{0}=0.0009765625$
(b) The probability that all bits are 0s is given by $P(X = 0)$. Using the binomial probability formula, we have:
$P(X=0)=(\frac{10}{0})(0.5)^{0}(1-0.5)^{10}=0.0009765625$
(c) The probability that exactly 5 bits are 1s and 5 bits are 0s is given by $P(X = 5)$. Using the binomial probability formula, we have:
$P(X=5)=(\frac{10}{5})(0.5)^{5}(1-0.5)^{5}=0.24609375$.
Therefore, the probability that exactly 5 bits are 1s and 5 bits are 0s is approximately $ 0.24609375$ or $24.609375$%. Place your bets accordingly!
