Question

A gambling book recommends the following “winning strategy” for the game of roulette: Bet $1 on red.

Probability
ANSWERED
asked 2021-08-23

A gambling book recommends the following “winning strategy” for the game of roulette:
Bet $1 on red. If red appears (which has probability \(\displaystyle\frac{{18}}{{38}}\)),then take the $1 profit and quit.
If red does not appear and you lose this bet ( which has probability \(\displaystyle\frac{{20}}{{38}}\) of occurring), make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit.
Let X denote your winnings when you quit.
(a) Find P{X>0}.
(b) Are you convinced that the strategy is indeed a “winning” strategy?
(c) Find E[X].

Expert Answers (1)

2021-08-24
(a)
Observe that possible outcomes of X are 1,-1,-3.
The gambler will win 1$ if he wins in the first bet or he loses the first bet, but he have success in the second and the third.
He will win -1$ if he loses the first and the second bet, but wins the third or loses the first, but wins the second.
Finally, he will win -3$ if he loses all the bets. Hence \(\displaystyle{P}{\left({X}{>}{0}\right)}={P}{\left({X}={1}\right)}=\frac{{18}}{{38}}+\frac{{20}}{{38}}\cdot{\left(\frac{{18}}{{38}}\right)}^{{2}}=\frac{{4059}}{{6859}}\approx{0.5918}\)
(b)
No, the strategy is not the winning one.
Even though that the probability that the gambler wins in this system is greater that 1/2, we will show in (c) that the expected winnings is less than zero since there is a big risk of losing all the bucks.
(c)
See that
\(\displaystyle{P}{\left({X}=-{1}\right)}={2}{\left(\frac{{20}}{{38}}\right)}^{{2}}\cdot\frac{{18}}{{38}}=\frac{{1800}}{{6859}}\)
\(\displaystyle{P}{\left({X}=-{3}\right)}={\left(\frac{{20}}{{38}}\right)}^{{3}}=\frac{{1000}}{{6859}}\)
so, using definition of expectation, we nave that
\(\displaystyle{E}{\left({X}\right)}={1}\cdot\frac{{4059}}{{6859}}-{1}\cdot\frac{{1800}}{{6859}}-{3}\cdot\frac{{1000}}{{6859}}\approx-{0.11}\)
which says that the gambler is expected to lose 11 cents.
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