1992 AHSME Problems/Problem 29

Problem

An unfair coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?

$\text{(A) } 25\bigg(\frac{2}{3}\bigg)^{50}\quad \text{(B) } \frac{1}{2}\bigg(1-\frac{1}{3^{50}}\bigg)\quad \text{(C) } \frac{1}{2}\quad \text{(D) } \frac{1}{2}\bigg(1+\frac{1}{3^{50}}\bigg)\quad \text{(E) } \frac{2}{3}$

Solution

Doing casework on the number of heads (0 heads, 2 heads, 4 heads...), we get the equation \[P=\left(\frac{1}{3} \right)^{50}+\binom{50}{2}\left(\frac{2}{3} \right)^{2}\left(\frac{1}{3} \right)^{48}+\dots+\left(\frac{2}{3} \right)^{50}\] This is essentially the expansion of $\left(\frac{2}{3}+\frac{1}{3} \right)^{50}$ but without the odd power terms. To get rid of the odd power terms in $\left(\frac{2}{3}+\frac{1}{3} \right)^{50}$, we add $\left(\frac{2}{3}-\frac{1}{3} \right)^{50}$ and then divide by $2$ because the even power terms that were not canceled were expressed twice. Thus, we have \[P=\frac{1}{2}\cdot\left(\left(\frac{1}{3}+\frac{2}{3} \right)^{50}+\left(\frac{2}{3}-\frac{1}{3} \right)^{50} \right)\] Or \[\frac{1}{2}\left(1+\left(\frac{1}{3} \right)^{50} \right)\] which is equivalent to answer choice $\fbox{D}$.

Solution 2

Denote $P_n$ as the probability that the number of heads is even. Then we have two cases:

$\bold{Case 1}$: (There are an odd number of heads for $(n-1)$ coin flips). Well to solve this case, we first ask the question; "What is the probability we get an odd number of heads for $(n-1)$ flips?" That is precisely $1-P_{n-1}$. Now, since we want an even number of heads, what is the probability that on the $n-th$ coin flip we get a head? $\frac{2}{3}$. So we have $\frac{2}{3}(1-P_{n-1})$.

$\bold{Case 2}$: (There are an even number of heads for $(n-1)$ coin flips). Since we now already have an even number of heads, we now want to flip and get a tail, or else the number of heads for $n$ flips will be odd. :/ So what is the probability we get a tail? $1-\frac{2}{3}=\frac{1}{3}$. Thus we have $\frac{1}{3}(P_{n-1})$.

Adding both cases, we have that $P_n=\frac{2}{3}-\frac{1}{3}P_{n-1}$. Now doing some tedious casework because wolfram alpha cannot find a pattern, we get the same answer as in $\boxed {D)}$.

Sorry for the very hand-wavy and non-rigorous answer! At least mine has a cool recurrence relation! :D

(Or just come up with the general formula $P_n=\frac{1}{2}(1+\frac{1}{3^n})$ Simple observation really. If you really want to get mathy, prove this through induction).

~th1nq3r

See also

1992 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 28
Followed by
Problem 30
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