Respuesta :
We are given a binomial probability distribution problem.
Probability of success: p = 50.8% = 0.508
Number of trials: n = 10 people
Fewer than 6 means x = 0, 1, 2, 3, 4, 5
Recall that the binomial probability distribution is given by
[tex]P(x)=^nC_x\cdot p^x\cdot(1-p)^{n-x}[/tex]Where nCx is the number of possible combinations.
[tex]P(x<6)=P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)[/tex]For p(x = 0):
[tex]\begin{gathered} P(x=0)=^{10}C_0\cdot0.508^0\cdot(1-0.508)^{10-0} \\ P(x=0)=0.00083 \end{gathered}[/tex]For p(x = 1):
[tex]\begin{gathered} P(x=1)=^{10}C_1\cdot0.508^1\cdot(1-0.508)^{10-1} \\ P(x=1)=10_{}\cdot0.508^1\cdot(0.492)^9 \\ P(x=1)=0.00858 \end{gathered}[/tex]For p(x = 2):
[tex]\begin{gathered} P(x=2)=^{10}C_2\cdot0.508^2\cdot(1-0.508)^{10-2} \\ P(x=2)=45\cdot0.508^2\cdot(0.492)^8 \\ P(x=2)=0.03987 \end{gathered}[/tex]For p(x = 3):0
[tex]\begin{gathered} P(x=3)=^{10}C_3\cdot0.508^3\cdot(1-0.508)^{10-3} \\ P(x=3)=120_{}\cdot0.508^3\cdot(0.492)^7 \\ P(x=3)=0.10978 \end{gathered}[/tex]For p(x = 4):
[tex]\begin{gathered} P(x=4)=^{10}C_4\cdot0.508^4\cdot(1-0.508)^{10-4} \\ P(x=4)=210\cdot0.508^4\cdot(0.492)^6 \\ P(x=4)=0.19836 \end{gathered}[/tex]For p(x = 5):
[tex]\begin{gathered} P(x=5)=^{10}C_5\cdot0.508^5\cdot(1-0.508)^{10-5} \\ P(x=5)=252\cdot0.508^5\cdot(0.492)^5 \\ P(x=5)=0.24578 \end{gathered}[/tex]Finally, the probability is
[tex]\begin{gathered} P(x<6)=P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5) \\ P(x<6)=0.00083+0.00858+0.03987+0.10978+0.19836+0.24578 \\ P(x<6)=0.6032_{}_{} \end{gathered}[/tex]There is a 0.6032 (60.32%) probability that fewer than 6 of those people are female.
