I was wandering aimlessly round a county fair not long ago, and came across a stall with a game I remember playing as a teenager, many years ago back in the UK – and losing. [The image here is not the one I saw; I just pulled it from the Web for illustration.]
You pay an entry fee and are given three darts. The goal is (and this may vary, though likely not by much) that if two of your three darts lands in a card, you win a small prize, and if all three do you win a more substantial prize.
I remember that as a teenager I was surprised, and disappointed, when none of my three throws ended up in a card. Not one. After all, at first glance it looks as though the cards occupy most of the board, so the odds should be in your favor.
Understanding what is going on provided yet another great example of number sense, the topic I focused on in my previous two posts. Let’s do some quick calculations. As usual, they don’t have to be accurate; we can make simplifications.
There are 4 rows of 5 cards, so 20 cards in all. According to a quick google search, playing-cards typically measure around 2.5 in by 3.5 in. Based on that information, a brief glance at the photo shows that the cards are placed on the board with a vertical and horizontal separation of roughly 1.25 in, with an outer border roughly the same size. So we can conclude that the dimensions of the board are
[5 x 2.5 + 6 x 1.25 = 12.5 + 7. 5 = 20] in wide by [4 x 3.5 + 5 x 1.25 = 14 + 6.25 = 20.25] in high.
Which means the total target area is 20 x 20.25 = 405 sq in.
The area of a single card is 2.5 x 3. 5 = 8.75 sq in, so the total area occupied by the cards is
[20 x 8.75 = 175] sq in.
Hence, the proportion of the target area occupied by a card is 175/405, or approximately 0.43.
That’s less than half, so the odds are against you on each throw. Assuming your throwing is random, which for most of us is will be, your throws will fail to land on card 57% of the time.
The probability of getting all three darts to land on cards to win a “substantial” prize is 0.433, which is approximately 0.08. You will win a real prize only 8 times in 100 attempts.
In fact, the odds are against you even winning a small prize, since the probability of two darts landing on a card is 0.432, or approximately 0.2, so you fail to win anything at all on 4 of every 5 attempts.
So the game is fine for a bit of fun, where you play maybe one or two rounds. Especially if the stall is there to raise money for a worthy charity. But for most of us, this is definitely not financially wise.
As for skilled darts players, in a commercial setting, the stallholder will surely stop anyone playing once they recognize the contestant’s skill (as happens in casinos with blackjack), so even if you play well, the odds are still heavily stacked against you making a killing.