SOLUTION
(a) The probability of at least 4 girls means the probability of getting 4 girls or the probability of getting 5 girls.
Using binomial probability formula
[tex]^nC_x\times p^x\times q^{n-x}[/tex]
Where p is the probability of success,
q = probability of failure
n = number of outcomes
x = number of successful events.
Probability of getting 4 girls means 4 success (4 girls) and 1 failure (1 boy)
So,
[tex]\begin{gathered} ^nC_x\times p^x\times q^{n-x} \\ ^5C_4\times(0.5)^4\times(0.5)^{5-4} \\ ^5C_4\times(0.5)^4\times(0.5)^1 \\ =0.15625 \end{gathered}[/tex]
Probability of getting 5 girls means all 5 success (5 girls) and 0 failure (0 boy)
So, we have
[tex]\begin{gathered} ^5C_5\times(0.5)^5\times(0.5)^{5-5} \\ ^5C_5\times(0.5)^5\times(0.5)^0 \\ =0.03125 \end{gathered}[/tex]
So, The probability of at least 4 girls becomes
[tex]\begin{gathered} 0.15625+0.03125 \\ =0.1875 \end{gathered}[/tex]
Therefore, the answer is 0.1875
(b) Probability of at most 4 girls is 1 - the probability of 5 girls
[tex]P(at\text{ most 4 girls) = 1 - P(5 girls) }[/tex]
P(5 girls) = 0.03125
So
[tex]\begin{gathered} P(at\text{ most 4 girls) = 1 - P(5 girls) } \\ P(at\text{ most 4 girls) = 1 - }0.03125 \\ =0.96875 \end{gathered}[/tex]
Therefore, the answer is 0.9688 to four decimal places
