# Help with a statistics question

I’m poor at the mechanics of figuring odds. I might be able to answer this with several hours of work, but I’m hoping someone else can do the math quickly and give me the answer.

There’s a board game where the players are trying to shoot down their opponent’s plane. You have four dice with the numbers 1-1-2-2-3-3. Every time your plane scores a hit, you roll the four dice to determine what area of your opponent’s plane was hit.

4 = cockpit
5,6 = tail
7,8,9 = wing
10,11 = engine
12 = fuel tank

There are two types of planes. F22’s are killed if they take one hit to the cockpit or fuel tank, two hits to the engine or tail, or four hits to the wings. F35’s are killed if they take one hit to the cockpit or fuel tank, two hits to the engine or tail, or three hits to the wings.

On average, how many hits will it take to shoot down a plane?

The chances of a hit to each part of the plane are as follows:

Cockpit: 1/81
Tail: 14/81
Wing: 51/81
Engine: 14/81
Fuel Tank: 1/81

Using tree diagrams, I’ve worked out the probabilities of the F35 getting shot down at each hit. Unfortunately, I can’t think of a simple way to type out the tree diagrams, so I’ll just give you the results.

Hit 1: 2.50%
Hit 2: 8.38%
Hit 3: 36.75%
Hit 4: 38.19%
Hit 5: 14.21%

So I multiplied the probability by the turn number, to get the average number of hits.

The average number of hits for an F35 is 3.53.

I’ll post the F22 stats in 15 minutes or so, if someone else doesn’t in the meantime.

For the F35, I get 3.5
I’ll get back to you with the F22

I get 4.2 for the F22, but I stuffed up my statistics a little so give or take 0.1.

Not that it matters or anything, but the chance of shooting down a plane on the first hit is 2.46%. It seems customary to nitpick everything in sight round these parts, so I thought I’d nitpick myself for a change.

Thank you both.