An unenlightened gambler

(a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds occur and bets

heavily on black at the next spin. Asked why he explains that black is “due by the law of averages.” Explain to the gambler what is wrong with this reasoning.

(b) After hearing you explain why red and black are still equally likely after five reds on the roulette wheel, the gambler moves to a poker game. He is dealt five straight red cards. He remembers what you said and assumes that the next card dealt in the same hand is equally likely to be red or black. Is the gambler right or wrong, and why?

On each spin of a roulette wheel, the red and black are equally likely to appear.

We can't foresee the outcomes of a chance process, yet they have a regular distribution over a large number of repetitions. According to the law of large numbers, the fraction of times a specific event occurs in numerous repetitions approaches a single number. The likelihood of a chance outcome is its long-run relative frequency. A probability is a number between $0$ (never happens) and $1$ (happens frequently) (always occurs).

When the gambler sees five reds in a row, he bets heavily on black on the next spin. After that, the gambler said that black is "due to the rule of averages." The gambler was incorrect in his logic because a spin is not affected by previous spins, and thus a spin after five consecutive spins will still have an equal chance of spinning red or black.

The player switches to a poker game after five reds on the roulette wheel. He is given five red cards in a row. On the same hand, the following card dealt is equally likely to be red or black.

Following the explanation, the gambler assumes that the following card dealt in the same hand will be red or black equally likely. However, the gambler was mistaken in his supposition. Because the five red cards that were dealt are no longer in the deck of cards, the gambler is incorrect. As a result, the deck of cards has more black cards than red cards, increasing the chances of being dealt a black card.