A fireworks display launches 40 fireworks in sequence. Each firework reaches a random height, independent of the height that any other firework reaches. Every time you see a firework that shoots higher than all of the ones that preceded it, you gasp. This means that you always gasp on the first firework. Is it more likely that your second gasp occurs on the seventh firework or that your final gasp occurs on the sixth firework?
The fireworks are all equally likely to reach any height, but no fireworks reach the same height as another.
It is more likely that your final gasp occurs on the sixth firework than that your second gasp occurs on the seventh firework.
Even though the fireworks reach random heights as they are launched in sequence, the math is the same if we imagine that the heights of the fireworks are picked in advance and then shown to you one at a time. The final gasp occurs at the firework that reaches the highest height. Since the 40 heights are random and independent of one another, every firework is equally likely to be the one that goes the highest. That means the sixth firework has a 1/40 chance of being the highest, and therefore your last gasp.
In general, the chance that any given firework is the highest out of the first n fireworks is simply 1/n. We know the first gasp occurs on the first firework. For the second gasp to occur on the seventh firework, two things must both be true:
The seventh firework is the highest of the first seven
The first firework is the highest of the first six
The first must be true because otherwise you won’t gasp on the seventh firework. The second must be true because otherwise you would have gasped a second time before reaching the seventh firework.
The probability of the first is 1/7 and the probability of the second is 1/6. The two events do not depend on each other, so the probability that both occur is (1/7) × (1/6) = 1/42.
Since 1/42 is smaller than 1/40, you are more likely to have your final gasp on the sixth firework than your second gasp on the seventh firework.
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