Answer:
[tex]1-(1-p^n)^m[/tex]
Step-by-step explanation:
For a coin, the probability of head showing in a single toss is [tex]p[/tex].
[tex]P(H)=p[/tex]
Its complement, the probability of not head is
[tex]P(\Sim H)=1-p[/tex]
This is a binomial distribution. In [tex]n[/tex] tosses, the probability of having all heads (i.e. [tex]n[/tex] heads) is
[tex]P(\text{all heads})=\binom{n}{n}p^n(1-p)^0=p^n[/tex]
Let's call this value [tex]a[/tex].
For [tex]m[/tex] coins, we determine the probability of at least 1 coin showing all heads by first finding its complement i.e. the probability of no coin showing all heads. This is also a binomial distribution.
[tex]P(\text{no coin showing all heads})=\binom{m}{0}a^0(1-a)^m=(1-a)^m[/tex]
[tex]P(\text{at least 1 coin showing all heads})=1-P(\text{no coin showing all heads})[/tex]
[tex]P(\text{no coin showing all heads})=1-(1-a)^m=1-(1-p^n)^m[/tex]
