Probability of at Least 2 Heads and 2 Tails in 5 Independent Coin Flips

[loli12]

There are 5 independent fair coin flips made,
What is the probabiltiy that at least two heads among the first 3 flips and at least two tails among the last three flips?

Except writing down all the possibilities, is there any other way to figure this out?

 

[HallsofIvy]

First think of it as two separate problems. What is the probability of "at least 2 heads in 3 flips"? What is the probability of "at least 2 tails in 3 flips"?

One thing to be careful about: the middle (third) flip is common to both!

I would do it this way:
a. Assume the middle flip is heads. What is the probability of getting "at least 1 head in the first 2 flips" (the middle head then gives you 2). Since the middle flip is heads you now need to answer "what is the probability of getting at least 2 tails in 2 flips". Since they must both happen, and are independent, multiply the probabilities.

b. Assume the middle flip is tails. Now, to get 2 heads in the first 3 flips, you must get 2 heads in the first 2 flips. What is that probabilty? Since the middle flip is tails you also must find the probability of getting 1 tail in the last two flips. Again, multiply those. (Because of the symmetry, don't be surprised if you get the same answer as in a.)

Since either a or b must happen, and they are mutually exclusive, the probability you want is the sum of those two.

 

[loli12]

thanks a lot!
this is a much better method!
I have never thought of setting the third flip to be a set outcome..
Thanks!