Question

a fair coin is tossed 8 times. find the probability of atleast five heads and at most five heads

Answer 1

The maximum number of combinations of heads and tails is 256, ie 2^8.

We can use Binomial coefficients or nCr formulae to work out how many cases have a particular number of heads.

nCr is the number of ways you can choose r objects from n objects without replacement and is given by n!/((n-r)! x r!) so to take a trivial example if we want to know how many ways we can have 1 head in 8 tosses we have n=8, r=1 and just substitute in to get 8!/(7! x 1!) = 8 and we can see that there are 8 ways of picking one object from 8 objects, so if we make a list

0 heads can happen 1 way
1 head can happen 8 ways
2 heads can happen 28 ways
3 heads can happen 56 ways
4 heads can happen 70 ways
5 heads can happen 56 ways
6 heads can happen 28 ways
7 heads can happen 8 ways
8 heads can happen 1 way

so to answer your questions for at least 5 heads we need to add the results for 5, 6, 7 and 8 above = 1 + 8 + 28 +56 = 93 so the probability of at least 5 heads is 93/256.

For the other part, at most 5 heads we add the results for 0, 1, 2, 3, 4 and 5 giving 225 so the probability of getting at most 5 heads is 225/256.

Answer 2

Answer:

Step-by-step explanation:

The maximum number of combinations of heads and tails is 256, ie 2^8.

We can use Binomial coefficients or nCr formulae to work out how many cases have a particular number of heads.

nCr is the number of ways you can choose r objects from n objects without replacement and is given by n!/((n-r)! x r!) so to take a trivial example if we want to know how many ways we can have 1 head in 8 tosses we have n=8, r=1 and just substitute in to get 8!/(7! x 1!) = 8 and we can see that there are 8 ways of picking one object from 8 objects, so if we make a list

0 heads can happen 1 way

1 head can happen 8 ways

2 heads can happen 28 ways

3 heads can happen 56 ways

4 heads can happen 70 ways

5 heads can happen 56 ways

6 heads can happen 28 ways

7 heads can happen 8 ways

8 heads can happen 1 way

so to answer your questions for at least 5 heads we need to add the results for 5, 6, 7 and 8 above = 1 + 8 + 28 +56 = 93 so the probability of at least 5 heads is 93/256.

For the other part, at most 5 heads we add the results for 0, 1, 2, 3, 4 and 5 giving 225 so the probability of getting at most 5 heads is 225/256.