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Module 8 (M8) – Handling data - Probability

Part of MathsM8: Handling data

This module builds on M7 Probability.

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Events that are not independent

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

If the probability of one event happening affects the probability of other events happening, then the two events are not independent.

Example

A sock drawer contains 5 white socks and 4 black socks. A sock is taken at random and put on. Another sock is taken and put on. What are the probabilities of the different sock pairings?

In this example, a sock is taken and not replaced in the drawer. This means that the next time a sock is picked, one of the socks will be missing. This will affect the remaining probabilities.

In this tree diagram, a white sock has been selected and put on, so there are no longer 5 white socks in the drawer. There are only 4. The total number of socks has also gone from 9 to 8.

An image of a probability tree, showing the probabilities of picking pairs of white socks and black socks from a drawer. In the first pick, the probability of picking a pair of white socks is a 5 out of 9 chance, and a picking a pair of black socks is 4 out of 8. In the second pick, the amount of socks has been reduced by one, which means that all the probability is now out of 8. In the second pick, if a white pair has already been chosen it is 4 out of 8 chances to pick a white pair, and 4 out of 8 to pick a black one. If a black pair has already been chosen, it is 5 out of 8 change the pair will be white, and 3 out of 8 chance of the pair being black.

So the probabilities of the different sock pairings are:

  • two white socks – \(\frac{5}{9} \times \frac{4}{8} = \frac{20}{72}\)

  • one white sock and one black sock – \(\frac{5}{9} \times \frac{4}{8} + \frac{4}{9} \times \frac{5}{8} = \frac{40}{72}\)

  • two black socks – \(\frac{4}{9} \times \frac{3}{8} = \frac{12}{72}\)

Question

Ross has a biscuit tin containing 10 wafer biscuits and 7 chocolate biscuits. He chooses one at random and eats it. He then chooses a second at random.

Complete the probability tree.

An uncompleted probability tree, showing Ross's probability of picking chocolate and wafer biscuits in two consecutive times.

Question

Using the completed probability tree from the previous question, find the probability that Ross chooses two biscuits that are the same.

Question

For a game of binary bingo, Ciarán has a bag containing 6 discs with a '0' on them and 4 with a '1' on them.

An image of ten equally shaped discs, with four yellow discs labelled '1' and six purple discs labelled '0'.

He chooses one at random and puts it on the table and then chooses a second also at random.

  • Draw a probability tree to represent his choices and use it to find the probability of choosing a 0 followed by a 1.

What is the probability of the two discs having different numbers?