Maths Games For Kids - The Possibilities Of Probability (part 2 Of 3)
In the introductory article we introduced the concept of probability being a mathematical measure of the likelihood of an event occurring
. This was illustrated with the tossing of a single coin, in which case the probability of it landing as a head is 1 in 2 (also expressed as 0.5) and the probability of it landing as a tail also 1 in 2 (0.5). When tossing a single coin, the possible outcomes are mutually exclusive - the coin cannot land as a head and a tail 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 are going to continue to look at coin tossing, but by introducing more that one coin we will significantly increase the complexity of mathematics required to calculate the probability of individual events.
First, take two 10 pence coins and throw them a few times, asking the kids to record the outcome of the throws. There appear to be three possible outcomes to tossing two coins: two heads, two tails or a head and a tail. However, swap one of the coins for a 50 pence coin and repeat the exercise, again asking the kids to record the results. There are now four possible outcomes: two head, two tails, the 10p as a head and the 50p as a tail, or finally the 10p as a tail and the 50p as a head. If one were to record the results as a grid, it would look like this:
10p 50p
H H
H T
T H
T T
By using two different coins, you reveal an additional outcome that using identical coins had concealed. When calculating probability, coin 1 being a head and coin 2 a tail is a different outcome to coin 1 being a tail and coin 2 a head, even if the two outcomes can't be distinguished visually. In the case of tossing two coins, one of the four outcomes is two heads, so the probability of this occurring is 1 in 4 (0.25). Similarly, the probability of throwing two tails is 1 in 4 (0.25). However, the probability of throwing a head and a tail is 2 in 4 (0.5), since two of the outcomes have one head and one tail, although it is a different coin which is the head in each case. Reassuringly, the sum of all the possible outcomes, 0.25 + 0.25 + 0.5, equals 1 as we would expect.
Probability may work as an abstract concept for kids, but the thing that really engages them is being shown practical applications for the subject.
The Odd Sock Problem
In this practical exercise, kids work out the probability of choosing a pair of socks of matching colour if they can't see the socks from which they have to choose. It mimics a real life problem that many blind people experience when getting dressed. Get one pair of red socks and one of green, separate them so there are four individual socks and put them in a bag. Next, get the kids to work out the probability that two socks drawn from the bag at random will make a matching pair.
There are two approaches to working out the probability in this case. The first involves gridding out all twelve possible outcomes and counting how many of the twelve include a matching pair. The second approach uses a logical short cut which says that the colour of the sock we draw first is immaterial, so long as we can calculate the probability that the second sock we draw is of a matching colour. It's worth pointing out that many kids will conclude that the sock 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 drawing both green socks.
In the concluding the article in the series we'll consider how the sock problem differs from the coin tossing scenario and work through the two approaches to calculating the probability of drawing matching socks from the bag. To reinforce the theoretical learning, the group can carry out a practical experiment to determine whether the actual results of drawing socks at random matches the predicted probability. Finally, we'll 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 picking a matching pair of socks.
by: Hannah McCarthy
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