What is the probability of heads in unfair coin when you flip the coin ten times?

Question

What is the probability of heads in unfair coin when you flip the coin ten times?

i came across this question and I haven't figured it out

Suppose that you flip an unfair coin ten times, where p(heads)=3/4 and p(tails)= 1/4, Find probability of

1.p(no heads)

2.p(exactly 9 heads)

3.p(exactly 7 heads)

4.p(at least 7 heads)

5.p(number of heads greater than number of tails)

i hop this help

Thanks

Answer 1

Have a look at the binomial distribution.

1. Probability of no heads: $$\bigg(\frac{1}{4}\bigg)^{10}$$
2. Exactly one head: $$\binom{10}{9}\,\bigg(\frac{1}{4}\bigg)\bigg(\frac{3}{4}\bigg)^9$$
3. Exactly seven heads: $$\binom{10}{7}\,\bigg(\frac{1}{4}\bigg)^3\bigg(\frac{3}{4}\bigg)^7$$
4. At least seven heads: $$\binom{10}{7}\,\bigg(\frac{1}{4}\bigg)^3\bigg(\frac{3}{4}\bigg)^7 + \binom{10}{8}\,\bigg(\frac{1}{4}\bigg)^2\bigg(\frac{3}{4}\bigg)^8 + \binom{10}{9}\,\bigg(\frac{1}{4}\bigg)\bigg(\frac{3}{4}\bigg)^9 + \bigg(\frac{3}{4}\bigg)^{10}$$
5. Number of heads greater than number of tails: same idea as in the previous case but starting at 6 heads.