Question

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 that $5$ consecutive reds have occurred and bets heavily on

black at the next spin. Asked why, he explains that “black is due.” Explain to the

gambler what is wrong with this reasoning.

b. After hearing you explain why red and black are still equally likely after $5$reds on the

roulette wheel, the gambler moves to a card game. He is dealt $5$straight red cards from

a standard deck with$26$red cards and $26$black 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.

Explain to the gambler what is wrong with this reasoning.

Short answer

a) Each spin is independent, receiving red and black has the same probability even after you know five reds have occurred.

b)No, because five red cards have already been eliminated from the card deck, leaving the deck with more black than red cards.

Step-by-step solution

Part (a) Step 1: Given information

We have to tell what is wrong with this reasoning.

Part (a) Step 2: Explanation

Because each spin is independent, receiving red and black has the same probability even after you know five reds have occurred. As a result, red has a $50\%$ probability of winning, and black has a $50\%$ chance of winning.

Part (b) Step 1: Given information

We have to explain to the gambler what is wrong with this reasoning.

Part (b) Step 2: Explanation

No, because five red cards have already been eliminated from the card deck, leaving the deck with more black than red cards.