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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 probability of an event occurring. This was illustrated with the toss of a single coin, in which case the probability of it landing heads is 1 in 2 (also expressed as 0.5) and the probability of it landing tails also 1 in 2 (0.5). When tossing a single coin, the possible outcomes are mutually exclusive – the coin cannot land heads and heads at the same time. The laws of probability state that the sum of the probabilities of each possible outcome must therefore be equal to 1.
In this second article of the series, we will continue to look at the coin toss, but by introducing more than one coin, we will greatly increase the complexity of the mathematics needed to calculate the probability of individual events.
First, take two 10 pence coins and toss them several times, asking the children to record the outcome of the tosses. There seem to be three possible outcomes to tossing two coins: two heads, two heads, or one head and one head. However, exchange one of the coins for a 50p coin and repeat the exercise, again asking the children to record the results. There are now four possible outcomes: two heads, two tails, 10p heads and 50p heads, or finally 10p heads and 50p heads. If we were to save the results as a grid, it would look like this:
10p – 50p
By using two different pieces, you reveal an additional result that the use of identical pieces had hidden. When calculating the probability, coin 1 being heads and coin 2 heads is a different outcome than coin 1 heads and coin 2 heads, even though the two outcomes cannot be visually distinguished. In the case of the two-coin toss, one of the four results is two heads, so the probability of this happening is 1 in 4 (0.25). Similarly, the probability of tossing two stacks is 1 in 4 (0.25). However, the probability of tossing heads and tails is 2 in 4 (0.5), since two of the results have heads and heads, although a different coin is the head in each case. Reassuringly, the sum of all possible outcomes, 0.25 + 0.25 + 0.5, equals 1 as expected.
Probability may work as an abstract concept for kids, but what really engages them is showing them practical applications for the topic.
The strange socks problem
In this hands-on exercise, children calculate the probability of choosing a matching color pair of socks if they do not see the socks to choose from. It mimics a real-life problem that many blind people encounter when getting dressed. Get a pair of red socks and a green one, separate them so that there are four individual socks and put them in a bag. Next, ask the children to calculate the probability that two randomly drawn socks from the bag form a matching pair.
There are two approaches to calculating the probability in this case. The first is to square the twelve possible outcomes and count how many of them include a matching pair. The second approach uses a logical shortcut that says the color of the sock we draw first is irrelevant, as long as we can calculate the probability that the second sock we draw is a matching color. It should be noted that many children will conclude that the sock problem is identical to the situation where two coins are tossed. However, there is an important difference between the two situations, which means that the probability of tossing two heads is not the same as pulling both green socks.
In the conclusion of the series article, we’ll look at how the socks problem differs from the coin toss scenario, and we’ll work through both approaches to calculating the probability of drawing matching socks from the bag. To reinforce theoretical learning, the group can perform a hands-on experiment to determine if the actual results of the random sock draw match the predicted probability. Finally, we will invite the group to use their knowledge of probability to explore if there are any strategies a blind person could use to increase their chances of choosing a matching pair of socks.
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