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Statistics and Probability - Complementary and Mutually Exclusive Events

Grade 8IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Probability is a measure of the likelihood of an event occurring, expressed as a value between 00 and 11 inclusive.

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Mutually Exclusive Events are events that cannot happen at the same time. For example, when flipping a coin, getting a 'Head' and a 'Tail' are mutually exclusive.

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The Addition Rule for Mutually Exclusive Events states that the probability of either event AA or event BB occurring is the sum of their individual probabilities.

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Complementary Events are two outcomes that are the only possibilities. If event AA happens, its complement A′A' (read as 'not AA') cannot happen. The sum of the probability of an event and its complement is always 11.

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In a sample space, if all outcomes are mutually exclusive and exhaustive (cover all possibilities), their probabilities must add up to 11.

📐Formulae

P(A)=Number of favorable outcomesTotal number of possible outcomesP(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

P(A or B)=P(A)+P(B) (for mutually exclusive events)P(A \text{ or } B) = P(A) + P(B) \text{ (for mutually exclusive events)}

P(A′)=1−P(A)P(A') = 1 - P(A)

P(A)+P(A′)=1P(A) + P(A') = 1

💡Examples

Problem 1:

A bag contains 55 red marbles, 33 blue marbles, and 22 yellow marbles. A marble is drawn at random. What is the probability that the marble is either red or yellow?

Solution:

The events 'drawing a red marble' and 'drawing a yellow marble' are mutually exclusive. Total marbles = 5+3+2=105 + 3 + 2 = 10. P(Red)=510=0.5P(\text{Red}) = \frac{5}{10} = 0.5 P(Yellow)=210=0.2P(\text{Yellow}) = \frac{2}{10} = 0.2 P(Red or Yellow)=P(Red)+P(Yellow)P(\text{Red or Yellow}) = P(\text{Red}) + P(\text{Yellow}) P(Red or Yellow)=0.5+0.2=0.7P(\text{Red or Yellow}) = 0.5 + 0.2 = 0.7

Explanation:

Since a single marble cannot be both red and yellow at the same time, we add the individual probabilities of the two mutually exclusive events.

Problem 2:

The probability of a certain football team winning their next match is 0.620.62. What is the probability that they do not win the match?

Solution:

Let WW be the event of winning. P(W)=0.62P(W) = 0.62. The event 'not winning' is the complement W′W'. P(W′)=1−P(W)P(W') = 1 - P(W) P(W′)=1−0.62=0.38P(W') = 1 - 0.62 = 0.38

Explanation:

Winning and not winning are complementary events. Their total probability must sum to 11. Therefore, we subtract the probability of winning from 11 to find the probability of the complement.

Problem 3:

A spinner has four sections: Green, Red, Blue, and White. The probability of landing on Green is 0.20.2, Red is 0.350.35, and Blue is 0.150.15. Calculate the probability of landing on White.

Solution:

The sum of all probabilities in the sample space must be 11. P(Green)+P(Red)+P(Blue)+P(White)=1P(\text{Green}) + P(\text{Red}) + P(\text{Blue}) + P(\text{White}) = 1 0.2+0.35+0.15+P(White)=10.2 + 0.35 + 0.15 + P(\text{White}) = 1 0.7+P(White)=10.7 + P(\text{White}) = 1 P(White)=1−0.7=0.3P(\text{White}) = 1 - 0.7 = 0.3

Explanation:

Since the four colors are the only possible outcomes and are mutually exclusive, we subtract the sum of the known probabilities from 11 to find the missing probability.