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Probability

Probability notation

How to find probability How to calculate probability Adding and subtracting decimals Adding and subtracting fractionsHere you will learn about mutually exclusive events, including what they are and how to find the probability of mutually exclusive events occurring.

Students will first learn about mutually exclusive events as part of statistics and probability in 7th grade.

**Mutually exclusive events** are two or more events that cannot occur at the same time. For example, getting heads and tails on a fair coin in a coin toss or rolling a 2 and a 3 on a six-sided die.

Mutually exclusive events are sometimes called disjoint events.

If two events are mutually exclusive then:

\textbf{P(A} ** or** \textbf{B) = P(A) + P(B)}

This means that the probability of event A or event B occurring is equal to the probability of event A occurring, plus the probability of event B occurring. This is known as the addition rule.

You need to consider other probability rules within probability theory if two or more events are not mutually exclusive (can happen at the same time). Other types of events include,

- Independent events
- Dependent events
- Complementary events
- Exhaustive events

How does this relate to 7th grade math?

**Grade 7 – Statistics and Probability (7.SP.C.8)**Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Use this quiz to check your grade 7 to 12 students’ understanding of probability. 15+ questions with answers covering a range of 7th to 12th grade probability topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 7 to 12 students’ understanding of probability. 15+ questions with answers covering a range of 7th to 12th grade probability topics to identify areas of strength and support!

DOWNLOAD FREEIn order to calculate the probability of mutually exclusive events:

**Confirm that the events are mutually exclusive.****Identify the probabilities of the events.****Add together the probabilities.**

A fair coin is flipped. Show that the probability of getting a heads or a tails is equal to 1.

**Confirm that the events are mutually exclusive.**

When flipping a coin, the two outcomes are heads or tails. These outcomes cannot occur at the same time and so the events are mutually exclusive.

2**Identify the probabilities of the events.**

The probability of a head is \cfrac{1}{2}.

The probability of a tail is \cfrac{1}{2}.

This is because the number of possible outcomes is 1 out of the total number of outcomes, 2 (heads or tails).

3**Add together the probabilities. **

The probability of getting a heads or a tails is 1.

Note: Such events are known as exhaustive as at least one of these events must occur when flipping a coin (heads or tails).

A card is selected at random from a deck. What is the probability of selecting a face card?

**Confirm that the events are mutually exclusive.**

As you are picking a face card from a deck, you can only pick one card at a time and so the events are mutually exclusive.

**Identify the probabilities of the events.**

There are 52 cards in a standard deck (excluding the jokers, as expected). There are four suits – Spades, Hearts, Diamonds, and Clubs. Within each suit are 3 face cards – Jack, Queen, and King.

This means that the probability of a Jack is \cfrac{4}{52}, the probability of a Queen is \cfrac{4}{52} and the probability of a King is \cfrac{4}{52}.

**Add together the probabilities.**

The probability of a face card is therefore the sum of the probabilities of a Jack, a Queen, or a King.

\cfrac{4}{52}+\cfrac{4}{52}+\cfrac{4}{52}=\cfrac{12}{52}

There are 3 red balls, 7 blue balls, and 5 green balls in a bag. One ball is picked from the bag. Calculate the probability of a red or a blue ball being picked.

**Confirm that the events are mutually exclusive.**

You cannot pick a red ball and a blue ball at the same time. Therefore, the events are mutually exclusive.

**Identify the probabilities of the events.**

The probability of picking a red ball is \cfrac{3}{15}.

The probability of picking a blue ball is \cfrac{7}{15}.

**Add together the probabilities.**

\cfrac{3}{15} + \cfrac{7}{15}=\cfrac{10}{15}

The probability of picking a red ball or a blue ball is \cfrac{10}{15} \, ( or \cfrac{2}{3}).

Note: It can be easier to work with fractional probabilities without simplifying them.

A card is drawn from a deck of cards. Calculate the probability of a 9, 10, or face card (Jack, Queen, or King) being picked.

**Confirm that the events are mutually exclusive.**

If you are picking one card, you cannot pick a 9, 10, and a face card at the same time, so the events are mutually exclusive.

**Identify the probabilities of the events.**

The probability of picking a 9 is \cfrac{4}{52}.

The probability of picking a 10 is \cfrac{4}{52}.

The probability of picking a face card is \cfrac{12}{52}.

**Add together the probabilities.**

The probability of a 9, 10, or face card (Jack, Queen, or King) being picked is

\cfrac{4}{52}+\cfrac{4}{52}+\cfrac{12}{52}=\cfrac{20}{52}.

A biased die is numbered 1-6. On this die, the probability of rolling a 6 is \cfrac{1}{3} and the probability of rolling a 5 or a 6 is \cfrac{1}{2}. Work out the probability of rolling a 5.

**Confirm that the events are mutually exclusive.**

You cannot roll a 5 and a 6 at the same time, so the events are mutually exclusive.

**Identify the probabilities of the events.**

The probability of rolling a 6 is \cfrac{1}{3}.

The probability of rolling a 5 or a 6 is \cfrac{1}{2}.

**Add together the probabilities.**

This time, you know the probability of rolling a 6 and the probability of rolling a 5 or 6.

If you write the probability of getting a 5 as P(5) then \cfrac{1}{3}+P(5)=\cfrac{1}{2}.

Solving this by subtracting \cfrac{1}{3} from both sides, you have

P(5)=\cfrac{1}{2}-\cfrac{1}{3}=\cfrac{3}{6}-\cfrac{2}{6}=\cfrac{1}{6}
.

The probability of rolling a 5 is \cfrac{1}{6}.

There are black, white, and blue socks only in a drawer. The probability of selecting a black sock from the drawer is 0.2 and the probability of selecting a white sock from the drawer is 0.45. What is the probability of selecting a blue sock?

**Confirm that the events are mutually exclusive.**

You cannot select a black, blue, and white sock at the same time, so the events are mutually exclusive.

**Identify the probabilities of the events.**

The probability of selecting a black sock is 0.2.

The probability of selecting a white sock is 0.45.

Since picking a black, white, or blue sock covers all possible outcomes, the events are exhaustive and therefore the total probability of selecting a black, a white, or a blue sock is 1.

**Add together the probabilities.**

If you call the probability of selecting a blue sock P(B) then 0.2+0.45+P(B)=1

Solving this, you have

\begin{aligned}
0.65+P(B)&=1\\\\
P(B)&=1-0.65\\\\
&=0.35
\end{aligned}

The probability of selecting a blue sock is 0.35.

- Use real-life examples of mutually exclusive events such as the outcomes of flipping a coin (heads or tails) or drawing different colored balls from a bag.

- Engage students with activities where they identify and classify events as mutually exclusive or not. This can involve group discussions, worksheets, quizzes, or problem-solving exercises.

- Use visual aids like Venn diagrams or probability tables to demonstrate mutually exclusive events. Show how their probabilities add up to 1 when considering all possible outcomes in the sample space.

- Highlight the difference between mutually exclusive and non-mutually exclusive events. For example, rolling an odd number and rolling a prime number on a die are not mutually exclusive.

**Adding fractions incorrectly**

Fractions can only be added if they have a common denominator.

**Applying**\textbf{P(A}**or**\textbf{B) = P(A) + P(B)}**to non mutually exclusive events**

For example, rolling a 3 and a prime number on a die. These events can happen at the same time since 3 is prime. Therefore they are not mutually exclusive and this formula does not work.

- Probability formula
- Independent events
- Probability scale
- Tree diagram probability
- Conditional probability
- Venn diagram

1. Which of these events are not mutually exclusive?

Rolling a 3 and a 4 on a die

Rolling an even number and a 5 on a die

Rolling an even number and a prime number on a die

Rolling an even number and a 3 on a die

Mutually exclusive events cannot happen at the same time.

2 is an even number and a prime number so it is possible to roll an even number and a prime number at the same time, therefore the events are not mutually exclusive.

2. Which of these events are mutually exclusive?

Drawing a red card and a 10 from a deck of cards

Drawing a black card and a diamond from a deck of cards

Drawing a black card and a 4 from a deck of cards

Drawing a diamond and an ace from a deck of cards

Mutually exclusive events cannot happen at the same time.

Diamonds are red cards, so you cannot draw a black card and a diamond at the same time.

3. In a bag there are 5 yellow balls, 6 blue balls, 7 red balls, and 8 green balls. One ball is picked. What is the probability that a yellow or red ball is picked?

\cfrac{12}{26}

\cfrac{5}{7}

\cfrac{14}{26}

\cfrac{13}{26}

There are 26 balls altogether.

The probability of picking a yellow ball is \cfrac{5}{26}.

The probability of picking a red ball is \cfrac{7}{26}.

The probability of picking a yellow or red ball is \cfrac{5}{26}+\cfrac{7}{26}=\cfrac{12}{26}.

4. What is the probability of the following spinner landing on a 1, 2 or 3?

\cfrac{6}{8}

\cfrac{3}{4}

\cfrac{7}{10}

\cfrac{3}{10}

The spinner has 10 sides.

The probability of landing on a 1 is \cfrac{4}{10}.

The probability of landing on a 2 is \cfrac{1}{10}.

The probability of landing on a 3 is \cfrac{2}{10}.

The probability of landing on a 1, 2 or 3 is \cfrac{4}{10}+\cfrac{1}{10}+\cfrac{2}{10}=\cfrac{7}{10}.

5. Lucy has a box of chocolates containing milk, white and dark chocolates.

The probability of picking a milk chocolate from the box is \cfrac{1}{2} and the probability of picking a milk chocolate or a white chocolate is \cfrac{4}{5}.

What is the probability of picking a white chocolate from the box?

\cfrac{13}{10}

\cfrac{3}{3}

\cfrac{3}{10}

\cfrac{7}{10}

The events are mutually exclusive therefore P(A or B) = P(A) + P(B).

\begin{aligned} \cfrac{4}{5}&=\cfrac{1}{2}+P(W)\\\\ \cfrac{4}{5}-\cfrac{1}{2}&=P(W)\\\\ P(W)&=\cfrac{8}{10}-\cfrac{5}{10}\\\\ &=\cfrac{3}{10} \end{aligned}

6. The probability that Hannah will be early to school is 0.3. The probability that Hannah will be on time to school is 0.5. What is the probability that Hannah will be late to school?

0.8

0.7

0.2

0.1

The events are mutually exclusive and the only possibilities are early, on time and late which means the sum of the probabilities must be 1.

\begin{aligned} 0.3+0.5+P(L)=1\\\\ P(L)=0.2 \end{aligned}

Mutually exclusive events are two or more events that cannot occur at the same time.

The addition rule for mutually exclusive events states that if two events A and B are mutually exclusive, then the probability of either event A or event B occurring is the sum of their individual probabilities: P(A or B)=P(A)+P(B).

This rule applies specifically to events that cannot occur simultaneously. In other words, if event A happening precludes event B from happening, and vice versa, they are mutually exclusive.

In probability theory, the sample space is the set of all possible outcomes of a random experiment.

- Compound events
- Probability distribution

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