Answer:
[tex]\dfrac{21}{32}=0.65625[/tex]
Step-by-step explanation:
If the probability of success is 50%, then p=0.5 and q=1-0.5=0.5.
At least three successes in six trials of a binomial experiment means that favorable are 3 successes, 4 successes, 5 successes and 6 successes.
1. 3 successes:
[tex]Pr_1=C^3_6p^3q^{6-3}=\dfrac{6!}{3!(6-3)!}\cdot (0.5)^3\cdot (0.5)^3=20\cdot \dfrac{1}{2^6}=\dfrac{5}{16}[/tex]
2. 4 successes:
[tex]Pr_2=C^4_6p^4q^{6-4}=\dfrac{6!}{4!(6-4)!}\cdot (0.5)^4\cdot (0.5)^2=15\cdot \dfrac{1}{2^6}=\dfrac{15}{64}[/tex]
3. 5 successes:
[tex]Pr_3=C^5_6p^5q^{6-5}=\dfrac{6!}{5!(6-5)!}\cdot (0.5)^5\cdot (0.5)^1=6\cdot \dfrac{1}{2^6}=\dfrac{3}{32}[/tex]
4. 6 successes:
[tex]Pr_4=C^6_6p^6q^{6-6}=\dfrac{6!}{6!(6-6)!}\cdot (0.5)^6\cdot (0.5)^1=1\cdot \dfrac{1}{2^6}=\dfrac{1}{64}[/tex]
Now, the probability of at least three successes in six trials of a binomial experiment is
[tex]Pr=Pr_1+Pr_2+Pr_3+Pr_4=\dfrac{5}{16}+\dfrac{15}{64}+\dfrac{3}{32}+\dfrac{1}{64}=\dfrac{20+15+6+1}{64}=\dfrac{42}{64}=\dfrac{21}{32}=0.65625[/tex]
