Roulette
In the game of roulette, the wheel has 38 slots numbered 0, 00, 1, 2, …, 36. A metal ball is spun around the wheel and can land in any of the slots. The slots numbered 0 and 00 are green, the odd numbers are red, and the even numbers are black..
d. Determine the probability that the metal ball lands in both the number 31 slot and a black slot at the same time. What term is used to describe this event?

Verified step by step guidance

1. Step 1: Understand the problem context. The roulette wheel has 38 slots: numbers 0, 00, and 1 through 36. The colors are assigned as follows: 0 and 00 are green, odd numbers are red, and even numbers are black.
2. Step 2: Identify the event of interest. We want the probability that the ball lands on the number 31 slot and also on a black slot at the same time.
3. Step 3: Analyze the event logically. Since the ball can only land in one slot at a time, it cannot simultaneously land on number 31 and any other slot. Therefore, the event of landing on 31 and landing on a black slot simultaneously is only possible if 31 is black.
4. Step 4: Determine the color of slot 31. Since 31 is an odd number, it is red, not black. Therefore, the event 'landing on 31 and landing on a black slot at the same time' cannot happen.
5. Step 5: Conclude the probability and terminology. The probability of this event is zero because it is impossible. This type of event, which cannot occur, is called an 'impossible event' in probability theory.

Key Concepts

Probability of an Event

Probability measures the likelihood of a specific outcome occurring, expressed as a number between 0 and 1. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of landing on a specific slot in roulette is 1 out of 38.

Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot. In the roulette question, landing on slot 31 and landing on a black slot simultaneously is impossible if slot 31 is not black, making these events mutually exclusive.

Intersection of Events

The intersection of two events refers to the event where both occur simultaneously. It is denoted as A ∩ B and its probability is found by considering outcomes common to both events. If no common outcomes exist, the intersection probability is zero.