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



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  • $\begingroup$ Please keep in mind that this is not a free homework solution service. $\endgroup$ – Aditya Dua Nov 25 '18 at 7:18

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.

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