91Ó°ÊÓ

In probability, events are said to be independent if the occurrence of one event does not affect the occurrence of another. For example, when Ralph bowls, whether or not he gets a strike in his first roll does not change the probability of getting a strike in the next roll.
This is the key idea behind independent events. Each roll is its own event and does not depend on previous rolls.
When calculating probabilities for independent events happening in sequence, we multiply the probabilities of each single event. For instance, the chance of getting two strikes in a row would be calculated as 0.30 (the probability of getting one strike) times 0.30 (the probability of getting another strike), resulting in a probability of 0.09.
\(... \text{P(Two Strikes)} = P(S) \times P(S) = 0.30 \times 0.30 = 0.09 \).
This multiplication rule is a fundamental principle when dealing with independent events.

In probability, complementary events are pairs of outcomes where one event occurs if and only if the other event does not. For example, if Ralph does not get a strike in a roll, it means he has missed the strike.
The probability of a complementary event happening can be found by subtracting the original event's probability from 1. For instance, if Ralph has a 0.30 probability of bowling a strike, then the probability of not getting a strike, also known as the complement, is \(... \text{P(No Strike)} = 1 - P(S) = 1 - 0.30 = 0.70 \).
Complementary events are useful when we want to determine the probability of an event not happening. This concept was applied in the solution to find the probability of Ralph not getting a clover but getting a turkey by calculating the probability of getting three strikes (turkey) and the probability of not getting the fourth strike (0.70).

The probability multiplication rule is used to find the probability of multiple independent events occurring in sequence. This rule states that to find the overall probability of all events happening, you multiply the probabilities of each individual event.
For example, if we want to calculate the probability that Ralph gets a turkey (three strikes in a row), we use the multiplication rule:
\(... \text{P(Turkey)} = P(S) \times P(S) \times P(S) = 0.30 \times 0.30 \times 0.30 = 0.027 \).
When Ralph aims for a clover (four strikes in a row), we continue applying the multiplication rule:
\(... \text{P(Clover)} = P(S) \times P(S) \times P(S) \times P(S) = 0.30 \times 0.30 \times 0.30 \times 0.30 = 0.0081 \).
We also use this rule in combination with our knowledge of complementary events to find more complex probabilities, like the probability of Ralph getting a turkey and missing the fourth strike. We found:
\(... \text{P(Turkey No Clover)} = P(Turkey) \times P(No Strike) = 0.027 \times 0.70 = 0.0189 \).