When I swing at a nail, I drive it all the way in with probability 1/2. With probability 1/4, I hit it half-way in, and with 1/4 probability I miss it entirely. Hitting it half-way in twice will drive it all of the way in. I'm pretty sure that if I swing 4 times at a nail, I'll get it all the way in almost all the time. Let's see if I'm right.
How many sequences of 4 swings could leave the nail still not knocked all the way in?
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When I swing at a nail, I drive it all the way in with probability 1/2. With probability 1/4, I hit it half-way in, and with 1/4 probability I miss it entirely. I'm pretty sure that if I swing 4 times at a nail, I'll get it all the way in almost all the time. Let's see if I'm right.
What is the probability I fail to get the nail driven completely in?
you could miss the nail 4 times - that's one sequence
you might hit it only once in the 4 swings - that's 4 sequences
that's a total of 5
probability you miss 4 times = (1/4)^4
P(hitting first time then miss next 3) = (1/4)(1/4)^3 = (1/4)^4
there are 4 different ways of doing this
which gives a probability of 4*(1/4)^4
So P(failing to drive the nail in) = 5*(1/4)^4 = 5/256