Problem 67 - Precalculus with Limits

Question

American roulette is a game in which a wheel turns on a spindle and is divided into \( 38 \) pockets.Thirty-six of the pockets are numbered \( 1-36 \), of which half are red and half are black. Two of the pockets are green and are numbered \( 0 \) and \( 00 \) (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number \( 00 \) pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number \( 14 \) pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins.

Step-by-Step Solution

Answer

(a) The probability of landing in the number 00 pocket is \(1/38\). (b) The probability of landing in a red pocket is \(9/19\). (c) The probability of landing in a green pocket or a black pocket is \(10/19\). (d) The probability of landing in the number 14 pocket on two consecutive spins is \(1/1444\). (e) The probability of landing in a red pocket on three consecutive spins is \(729/6859\).

Step 1: Understand the question

The game of American Roulette has 38 equally likely outcomes. Therefore, the probability of any single outcome is \(1/38\) because each pocket has an equal chance be selected.

Step 2: Find the probability of landing in the number 00 pocket

There is only one '00' pocket, out of the total 38 pockets. Therefore, the probability is \(1/38\).

Step 3: Find the probability of landing in a red pocket

There are 18 red pockets out of a total 38 pockets. So, the probability of landing in a red pocket is \(18/38\) or \(9/19\).

Step 4: Find the probability of landing in a green pocket or a black pocket

A green pocket or a black pocket outcome means the ball could land in either of the two, so we add their individual probabilities. There are 2 green pockets and 18 black pockets, so the probability is \((2 + 18)/38\) or \(20/38 = 10/19\).

Step 5: Find the probability of landing in the number 14 pocket on two consecutive spins

The events of spinning the roulette wheel are independent, so the probability of landing in pocket 14 twice in a row is the product of their individual probabilities, which is \((1/38) * (1/38)\) or \(1/1444\).

Step 6: Find the probability of landing in a red pocket on three consecutive spins

Similarly, the probability of landing in a red pocket thrice in a row is the product of their individual probabilities, which is \((9/19) * (9/19) * (9/19)\) or \(729/6859\).