Maths Games For Kids - The Possibilities Of Probability (part 3 Of 3)
The previous article concluded with the setting of a probability problem for the group
. The challenge for them was to work out the probability of picking a matching pair of socks if two socks were picked from a bag containing two red socks and two green socks. The person doing the picking can't see the socks, either because it's a perfect peep-proof bag or because they're blindfolded, a situation which mimics the problem that a blind person may face if getting dressed unaided.
The first option for working out the probability is to grid out all the possible results of picking the socks. If each sock is treated separately (we've assigned the identifiers R1 and R2 to the red socks and G1 and G2 to the green socks) and the order in which the socks are drawn is taken in consideration, there are twelve possible permutations in which the socks can be drawn:
R1 R2 *
R1 G1
R1 G2
R2 G1
R2 G2
R2 R1 *
G2 R1
G2 R2
G2 G1 *
G1 R1
G1 R2
G1 G2 *
Four of the twelve possible permutations, marked with an asterisk on the grid, result in the picking of a pair of socks that match. The probability, therefore, of picking a matching pair of socks is 4 in 12, which we can simplify to 1 in 3 and express in decimal form as 0.33.
The second approach to calculating the probability uses a logical short cut which greatly speeds up the process. Using this approach, one can ignore which colour sock is first drawn from the bag and simply calculate the probability that the second sock drawn will be of the same colour. After the first sock has been drawn, there will be three socks left in the bag. One of the three socks will be the same colour as the sock already drawn and the remaining two will be of the other colour. The probability of drawing a matching pair is therefore 1 in 3, as you have to pick the one matching sock of the three remaining to make a pair.
Having established the probability of picking a matching pair of socks, now ask the group to calculate the probability of randomly picking the green pair of socks out of the bags. Kids who have used the grid approach will very quickly be able to determine that there are two outcomes of the twelve which result in the green pair of socks being chosen, so the probability of this is 2 in 12, which simplifies to 1 in 6. Those who have used the logical short cut approach for the earlier calculation will find that things become a little more complex. To pick a green pair of socks, the first sock that is picked must be green, the probability of which is 2 in 4 (I in 2) and the second sock must also be green, the probability of which is 1 in 3. To calculate the probability of both these outcomes occurring successively, as they must do to result in the green pair of socks being picked, one multiplies the probability of the first event occurring by the probability of the second event occurring:
1 in 2 x 1 in 3 = 1 in 6
which produces, as expected, the same result as the grid approach.
Having established the probability of drawing:
A matching pair of socks
The green pair of socks
allow each member of the group to have six goes at drawing socks from the bag and get them to record the results. Discuss as a group why an individual's results differ from the calculated probability. Next add up the individual results to demonstrate that the more times an event is repeated, the closer the actual outcomes should be to the calculated probability.
So, how does drawing two socks from four differ from the tossing of two coins? Clearly the two situations are different, as the probability of throwing two heads is 1 in 4, but the probability of drawing the green pair of socks is 1 in 6. When you're drawing socks from a bag, the removal of the first sock alters the number of socks remaining in the bag and the probability of picking the matching sock. If you drew the first sock, noted its colour and then put it back in the bag before drawing a second time, then you would be exactly mimicking the coin situation. In this case, the probability of picking a matching pair would be 1 in 2 and the probability of picking two green socks would be 1 in 4.
Finally, what strategies could a blind person use to increase the probability of picking matching socks? Strangely, if one of the socks gets lost, the probability of picking the pair from the three remaining socks is still 1 in 3. However, owning more socks is an effective way of increasing the chances of picking matching pairs. If there are 3 red and 3 green socks in the bag, the chances of picking a matching pair increases to 2 in 5 (0.4). With 50 red and 50 green socks, the probability becomes 49 in 99. Indeed, the more socks there are to choose from, the closer the chances of picking a matching pair of socks comes to 1 in 2 (0.5). This result will seem counter intuitive to many kids, which is why teaching them about probability is both fun and frustrating in equal measure.
by: Hannah McCarthy
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