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## Maths Games For Kids – The Possibilities of Probability (Part 2 of 3)

In the introductory article we introduced the concept of probability as a mathematical measure of the likelihood of an event occurring. This was illustrated by tossing a single coin, in which case the probability that it will land as heads is 1 in 2 (also expressed as 0.5) and the probability that it will land as tails is also 1 in 2 (0.5). When a single coin is tossed, the possible outcomes are mutually exclusive: the coin cannot land heads and tails at the same time. The laws of probability say that the sum of the probabilities of each possible outcome must therefore equal 1.

In this second article in the series, we will continue to look at coin tossing, but by introducing more than one coin we will significantly increase the complexity of the mathematics required to calculate the probability of individual events.

First, take two 10 cent coins and toss them a few times, asking the children to record the result of the tosses. There seem to be three possible outcomes for tossing two coins: two heads, two tails, or one head and one tail. However, change one of the coins to a 50 cent coin and repeat the exercise, again asking the children to record the results. There are now four possible outcomes: two heads, two tails, 10p as head and 50p as tail, or finally 10p as tail and 50p as head. If one were to record the results as a grid, it would look like this:

10p – 50p

HH

HT

TH

T-T

By using two different coins, you reveal an additional result that using identical coins had hidden. When calculating the probability that coin 1 is a head and coin 2 is a tail is a different outcome than coin 1 is a tail and coin 2 is a head, even if the two outcomes are visually indistinguishable. In the case of tossing two coins, one of the four outcomes is two heads, so the probability of this happening is 1 in 4 (0.25). Similarly, the probability of tossing two tails is 1 in 4 (0.25). However, the probability of tossing heads and tails is 2 in 4 (0.5) since two of the outcomes have a head and a tail, even though it is a different coin that is the head in each case. Reassuringly, the sum of all possible outcomes, 0.25 + 0.25 + 0.5, equals 1 as we might expect.

Probability can work as an abstract concept for children, but what really interests them is showing practical applications for the subject.

The strange sock problem

In this hands-on exercise, children calculate the probability of choosing a pair of matching colored socks if they cannot see the socks they have to choose from. It mimics a real-life problem that many blind people face when dressing. Take one pair of red socks and one pair of green socks, separate them so that there are four individual socks, and put them in a bag. Then have the children calculate the probability that two socks drawn at random from the bag will make a matching pair.

There are two approaches to calculating the probability in this case. The first involves ranking the twelve possible outcomes and counting how many of the twelve include a matching pair. The second approach uses a logical shortcut that says that the color of the sock we draw first is not important, as long as we can calculate the probability that the second sock we draw is the same color. It is worth noting that many children will conclude that the socks problem is identical to the situation where one is tossing two coins. However, there is an important difference between the two situations, which means that the probability of throwing two heads is not the same as getting both green socks.

At the conclusion of the article series, we will consider how the sock problem differs from the coin toss scenario and work with both approaches to calculate the probability of drawing matching socks from the bag. To reinforce the theoretical learning, the group can conduct a practical experiment to determine whether the actual results of drawing socks at random match the predicted probability. Finally, we will invite the group to use their knowledge of probability to explore whether there are any strategies that a blind person could use to increase their chances of choosing a matching pair of socks.

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