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To determine the probability that two individuals, each flipping a coin four times, end up with the same number of heads, we need to consider the binomial distribution. Each coin flip is an independent event with two possible outcomes: heads or tails, each with a probability of 0.5.
For each individual flipping a coin four times, the number of heads follows a binomial distribution with parameters n=4 (number of trials) and p=0.5 (probability of getting heads in each trial). The probability mass function (PMF) for a binomial distribution is given by:
P(X=k)=(kn)pk(1−p)n−k
Where:
For n=4 and p=0.5, the probabilities for different numbers of heads k (ranging from 0 to 4) are:
To find the probability that both individuals get the same number of heads, we sum the probabilities of both individuals having the same outcome for each possible number of heads:
P(same number of heads)=∑k=04P(X1=k)⋅P(X2=k)
Substituting in the probabilities calculated:
\begin{align*} P(\text{same number of heads}) &= \left(\frac{1}{16}\right)^2 + \left(\frac{4}{16}\right)^2 + \left(\frac{6}{16}\right)^2 + \left(\frac{4}{16}\right)^2 + \left(\frac{1}{16}\right)^2 \\ &= \frac{1}{256} + \frac{16}{256} + \frac{36}{256} + \frac{16}{256} + \frac{1}{256} \\ &= \frac{70}{256} \\ &= \frac{35}{128} \\ &\approx 0.2734 \end{align*}
Thus, the probability that both individuals end up with the same number of heads after flipping a coin four times is approximately 0.2734, or 27.34%.