Probability of Sharpshooter Missing After 3 Shots

[Tomp]

Question
An expert sharpshooter misses a target 10 percent of the time.

I have a question
"What is the probability that the first miss comes after the 3rd shot?"

Is this as simple as 0.9^3 * 0.1?

 

[Bacle2]

Look at the sample space :

{ HHHM, HHHHM,...}

 

[Simon Bridge]

Yep, the key word is "after".

 

[HallsofIvy]

The question is "does 'after'" mean "immediately after" or would five hits and then a miss be "the first miss is after the first three shots".

If you mean the first, then, yes, (.9)^3(.01) is correct.

If the second, then you need to expand that to larger numbers of initial hits:
(.9)^3(.01)+ (.9)^4(.01)+ (.9)^5(.01)+...

You might recognize that as part of a geometric sequence and so find a simple formula for sum.

 

[ssd]

The question is "does 'after'" mean "immediately after" or would five hits and then a miss be "the first miss is after the first three shots".

If you mean the first, then, yes, (.9)^3(.01) is correct.

If the second, then you need to expand that to larger numbers of initial hits:
(.9)^3(.01)+ (.9)^4(.01)+ (.9)^5(.01)+...

You might recognize that as part of a geometric sequence and so find a simple formula for sum.

Not 0.01 but 0.1 in all cases.:)

It is interesting to check, if we consider the 'after case' (not immediately after) then the asked event is nothing but "3 hits in first 3 shots" (whatever happens later does not matter). Therefore the answer will be (0.9)^3.