Q. 3.18 - A First Course in Probability

Question

Mr. Jones has devised a gambling system for winning at roulette. When he bets, he bets on red and places a bet only when the $10$previous spins of the roulette have landed on a black number. He reasons that his chance of winning is quite large because the probability of $11$ consecutive spins resulting in black is quite small. What do you think of this system?

Step-by-Step Solution

Answer

The wheel has no memory and the probability of win is $1/2.$

Step: 1 Probability of success:

The probability of success is $1/2.$

Although the chances of $11$ consecutive black numbers in roulette is low, when 10 consecutive black numbers have been witnessed, the result space only contains $10$ or more consecutive black numbers, making 11 a strong possibility.

$P\left(11\text{ consecutive black numbers }\right)$

Step: 2 probability of success:

The success rate is a statistical term that is widely utilised in the pharmaceutical business, as well as by health authorities to aid decision-making. The idea of success probability is directly connected to conditional power and predictive power.

Step: 3 Probability:

In actuality, he stakes his money on:

$P(11\text{ consecutive black numbers }\mid 10\text{ consecutive black numbers })=\frac{1}{2}$

Because the $11$th roll is independent of the previous ten, the chance of a red number is $1/2$as a consequence.

Another way to put it is that the roulette wheel has no "memory," that is, it flashes red with probability $1/2$regardless of the previous ten results.