Module 7 (M7) – Handling data - Probability

Part of MathsM7: Handling data

This module builds on M6 Probability.

Number of outcomes

To calculate the probability of an event, the total number of possible outcomes is often required. For simple situations, making a list or completing a sample space diagram is enough but in more complex situations the product rule for counting is needed.

The rule is:

To find the total number of outcomes for two or more events, multiply the number of outcomes for each event together.

Example

A restaurant menu offers 4 starters, 7 main courses and 3 different desserts. How many different three-course meals can be selected from the menu?

Solution

Multiplying together the number of choices for each course gives

\(4 \times 7 \times 3 = 84 \)

Eighty-four different three-course meals can be selected.

Question

Miley is choosing a new 4-digit pin number. Her friend says that if you allow zeros and repeated digits, there are 40 different 4-digit numbers available (0 to 9 in each of 4 slots so \(4 \times 10 = 40\)).

Miley says that there are a lot more than that.

Is Miley right and if so, how many different 4-digit numbers are there?

Mutually exclusive events

Two events are mutually exclusive if they cannot occur at the same time.

Example

Selecting an Ace

Selecting a Jack

An image of a deck of cards, spread out, with the aces of diamonds, hearts and clubs sitting at the top.

The 'OR' rule: the addition rule

If A and B are mutually exclusive events.

  • \(P(\text{A or B}) = P(A) + P(B)\)

Example

A bag of sweets contains 7 toffees, 5 gummies and 6 caramels.

A sweet is selected at random. What is the probability of a toffee or a caramel being selected?

Solution

The events 'choosing a toffee' and 'choosing a caramel' are mutually exclusive - they cannot happen at the same time when just one sweet is being selected.

Use the addition rule (the 'OR' rule):

  • \(P(\text{A or B}) = P(A) + P(B)\)

  • \(P(\text{toffee}) = \frac{7}{18}\) – there are 18 sweets altogether, and 7 are toffee

  • \(P(\text{caramel}) = \frac{6}{18}\)

  • \(P(\text{toffee OR caramel}) = \frac{7}{18} + \frac{6}{18} = \frac{13}{18}\)

Question

Lisa and Rosa are playing a game with animal cards. Lisa has 4 lion cards, 2 giraffe cards and 1 tiger card. Rosa takes one of Lisa's cards without looking at the pictures.

What is the probability that she takes a giraffe or a tiger card?

Independent events

Two events are independent if the occurrence of one event is unaffected by the occurrence of the other.

'Tossing a coin and getting a tail' and 'rolling a dice and getting a 6' are independent events.

The 'AND' rule: the multiplication rule

If A and B are independent events

\(P(\text{A and B}) = P(A) \times P(B)\)

Example

Rory rolls a dice and tosses a coin at the same time. What is the probability that he gets tails on the coin and a 6 on the dice?

Solution

Use the multiplication Rule (the AND rule):

  • \(P (\text{A and B}) = P(A) \times P(B)\)

  • \(P(\text{tails}) = \frac{1}{2}\)

  • \(P(\text{six}) = \frac{1}{6}\)

  • \(P(\text{tails and six}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}\)

Question

An image showing an illustration of spinner with four quarters, two labelled blue, one labelled purple and one labelled red, as well as an 8-sided die.

Hannah has a spinner with four equal sectors and an 8-sided dice as shown above.

She spins the spinner and rolls the dice at the same time.

What is the probability that she gets blue on the spinner and a 5 on the dice?

Tree diagrams

Tree diagrams are a visual way of showing all possible outcomes of two or more events. Each branch is a possible outcome and is labelled with a probability.

Two events are independent if the probability of the first event happening has no impact on the probability of the second event happening.

For example, the probability of rolling a 6 on a dice will not affect the probability of rolling a 6 the next time. The scores on the dice are independent.

If a dice was to be rolled twice, the tree diagram would look like this:

An image showing a tree diagram of the probability of rolling a 6 over two dice rolls. Roll 1 shows that there is a one in 6 chance of rolling a 6, rather than a 5 in 6 chance of not rolling a six. This is replicated twice on the second roll.

There are four possible outcomes. To work out the probabilities of each total outcome, multiply the probabilities together.

An image showing a tree diagram of the probability of rolling a 6 over two dice rolls. Roll 1 shows that there is a one in 6 chance of rolling a 6, rather than a 5 in 6 chance of not rolling a six. This is replicated twice on the second roll. The diagram shows the probabilities of each total outcome by multiplying the probabilities together. Getting two 6s is one over six multiplied by one over six, equaling 1 over 36. Getting a six, and not a six is one over six multiplied by five over six, equaling 5 over 36. Getting no sixes on either roll is five over six multiplied by five over six equaling 25 over 36.

Question

A bag contains 4 blue counters and 3 red counters. A box contains 5 blue counters and 2 red counters.

Complete the tree diagram and work out the probability of selecting two red counters.

Question

Zoë plays tennis with Helen every Saturday. The probability that Zoë wins is 0.6.

Complete the probability tree and use it to find

  • the probability that Zoë loses two Saturdays in a row
  • the probability that Zoë loses one of the games and wins the other.
An uncompleted probability tree, showing Zoë's probability of winning versus losing over two consecutive Saturdays.

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