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?

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.

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

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.

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.