Graham and Sally are playing a game where they each randomly place five counters in five different squares on a grid like the one shown. They then take turns to guess the position of each other’s counters by stating a grid reference, for example, E5. Part a) What is the probability that Graham guesses the position of one of Sally’s counters on his first guess?
There’s also a part b) that we’ll come on to. So to answer part a), what we’re trying to do is see what the probability is that Graham guesses the position of one of Sally’s counters on his first guess. And when we’re dealing with probability, we know that the probability of an outcome occurring is equal to the number of successful outcomes over the total number of outcomes.
So the first thing we look at is the total number of outcomes. In our situation, this is gonna be the total number of squares because the squares are the number of places that Graham could guess if he was trying to guess the position of one of Sally’s counters. Well, if we look at the grid, it’s a five-by-five grid. Well, therefore, there are 25 total number of squares. So that means our total number of outcomes is 25.
So now, what we need to do is look at the number of successful outcomes. Well, in our case, that is going to be the number of squares that are gonna have a counter on them. And we’re told that there are five counters in five different squares. So therefore, our number of successful outcomes is going to be five.
It’s worth noting though that the question does tell us that there are five different squares that the counters are placed upon because if five counters were placed on any squares and they could double up on the same square, then we would not have the same number of successful outcomes because we could conceivably have two, three, four, any number of our five counters on one square.
But as we said, as they’re on different squares, we’re gonna have five squares with counters on. So therefore, we can say the probability of choosing a square with a counter on is gonna be equal to five over 25 because as we said that was our number of successful outcomes over our total number of outcomes.
So now we see, can we simplify this any further? Well, yes, we can because we can divide the numerator and denominator by five because five is a factor of both five and 25. So this is gonna be equal to one over five or one-fifth.
Now, it’s worth noting at this point that if you’ve left your answer as five over 25, you probably would have got the mark because it hasn’t asked for it in its simplest form. However, it is good practice to simplify where you can. So that’s part a) finished. So now, let’s move on to part b).
So in part b), we’re told “In one game, Graham places all five of his counters in one diagonal line. Identify which square must contain a counter.”
Now, there are only two possible ways that Graham could have placed all five of his counters in a diagonal line. Now, this is the first way you could have done it and that’s from the top left corner to the bottom right corner. And the second is like this: from the bottom left to the top right. Okay, so we’ve shown the two ways that Graham can place his counters in the arrangement that he wants to.
So now, what we need to do is identify which square must contain a counter. Well, there is only one possible answer and it’s the square that I’ve circled. Because either way, so either way that he can put his counters in a diagonal line, they must go through this square. So that means if Graham places his counters from top left to bottom right or bottom left to top right, it always have a counter placed upon this square.
And the best way to identify this square is to use a grid reference as they’ve done in the example. And to do that, we need to recognize which column and which row the square is going to be in. So it lies in the column with C and the row with three. So therefore-so therefore, we can say that C3 must contain a counter.