Puzzle 18 of 26 · Easy

The gambler and the friend

A smaller version of the roulette gambler: a wheel has 6 equal sections, and a $1 bet on one section pays 5 to 1. You bet $1 on the same section for 6 spins. What’s the chance you end up behind?

Your section is one of 6.

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Answer: 15625/46656 ≈ 33.5%

Chance of each number of wins in 6 spins. Only 0 wins leaves you behind.

Step 1 · One win is enough to break even
$\displaystyle \text{one win}\colon + \$5; \text{ five losses}\colon - \$5$
A single win pays back exactly what the other 5 bets lose, so you’re only behind if you never win.
Step 2 · No win in 6 spins
$\displaystyle \left(\frac{5}{6}\right)^{6} = \frac{15625}{46656}$
Each spin misses your section with chance 5/6, and spins don’t affect each other, so multiply 6 times.

Answer $\displaystyle \frac{15625}{46656} \approx 33.5\%$

Step 2 of 2

The method · Unit 5, DistributionsBinomial distributionFind the chance of k wins in n tries

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