The formula for binomial probability is expressed as
P(x) = nCx * p^x * q^(n - x)
where
n = number of trials
x = number of successes
p = probability of success
q = probability of failure
From the information given,
n = 29
p = 0.185
q = 1 - p = 1 - 0.185 = 0.815
a) For the probability that you win 6 prices, x = 6
P(x = 6) = 29C6 * 0.185^6 * 0.815^(29 - 6)
P(x = 6) = 0.1723
b) For the probability that you win more than 8 prices, x = 8. From the binomial probability distribution calculator,
[tex]P(x>8)\text{ = 0.0729}[/tex]c) we would find the probability that you win more than 4 prizes and subtract it from the probability that you win 7 or lesser pizes. x = 4 and x = 7
From the binomial probability distribution calculator,
[tex]\begin{gathered} P(x>4)\text{ = 0}.6438 \\ P(x\text{ }\leq7)\text{ = 0.8468} \\ P(4\text{ }\leq\text{ x }\leq7)\text{ = 0.8468 - 0.6438 = 0}.203 \end{gathered}[/tex]d) For the probability that you win 3 prizes or fewer, x = 3. From the binomial probability distribution calculator,
[tex]P(x\leq3)\text{ = 0}.1890[/tex]