In the introductory article we introduced the concept of probability as a mathematical measure of the chance of an event occurring. This was illustrated by tossing a single coin, in which case the probability of heads up is 1 in 2 (also expressed as 0.5) and the probability of tails also being 1 in 2 (0.5). When tossing a single coin, the possible outcomes are mutually exclusive: the coin cannot land heads and tails at the same time. The laws of probability state that the sums of the probabilities of each possible outcome must therefore equal 1.
In this second article in the series, we will continue to discuss coin toss, 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 10p coins and toss them several times, ask the children to record the results of the tosses. It seems that there are three possible outcomes when tossing two coins: two heads, two tails, or heads and tails. 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 tails, or finally 10p tails and 50p heads. If one were to record the results as a grid, it would look like this:
10p-50p
H.H
HT
TH
TT
By using two different coins, you reveal an additional result that the use of identical coins had hidden. When calculating probability, if coin 1 is heads and coin 2 is tails is a different outcome than coin 1 is tails and coin 2 is heads, even if the two outcomes cannot be visually distinguished. In the case of flipping 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 getting two tails is 1 in 4 (0.25). However, the probability of getting heads and tails is 2 in 4 (0.5), since two of the outcomes have heads and tails, even though a different coin comes up heads in each case. Reassuringly, the sum of all possible outcomes, 0.25 + 0.25 + 0.5, equals 1, as you might expect.
Probability may work as an abstract concept for children, but what really engages them is being shown practical applications for the topic.
The Weird Socks Problem
In this hands-on exercise, children find the probability of choosing a pair of socks of the same color if they cannot see which socks they have to choose from. It mimics a real life problem that many blind people experience when getting dressed. Get one pair of red socks and one pair of green, separate them so there are four individual socks, and put them in a bag. Next, have the children find 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 gridding 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 irrelevant, 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 sock problem is identical to the two-coin toss situation. However, there is an important difference between the two situations which means that the probability of drawing two heads is not the same as drawing both green socks.
At the end of the series article, we’ll consider how the sock problem differs from the coin-toss scenario and discuss the two approaches to finding the probability of drawing matching socks from the bag. To reinforce theoretical learning, the group can conduct a hands-on experiment to determine if the actual results of randomly pulling socks match the predicted probability. Finally, we will invite the group to use their knowledge of probability to explore if there are any strategies that a blind person could use to increase their chances of choosing an identical pair of socks.