5. in the current nba regular season, the captain of the new york knicks, jalen brunson, has made…

Explanation:

Step1: Identify the binomial - distribution parameters

The binomial - distribution formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 7$, $p=0.4$, and $1 - p = 0.6$.

Step2: Calculate the probability for part (a)

The probability that he makes at least 5 shots is $P(X\geq5)=P(X = 5)+P(X = 6)+P(X = 7)$. $P(X = 5)=C(7,5)\times(0.4)^{5}\times(0.6)^{2}=\frac{7!}{5!(7 - 5)!}\times(0.4)^{5}\times(0.6)^{2}=21\times0.01024\times0.36 = 0.0774144$. $P(X = 6)=C(7,6)\times(0.4)^{6}\times(0.6)^{1}=\frac{7!}{6!(7 - 6)!}\times(0.4)^{6}\times(0.6)^{1}=7\times0.004096\times0.6 = 0.0172032$. $P(X = 7)=C(7,7)\times(0.4)^{7}\times(0.6)^{0}=\frac{7!}{7!(7 - 7)!}\times(0.4)^{7}\times1=1\times0.0016384\times1 = 0.0016384$. $P(X\geq5)=0.0774144 + 0.0172032+0.0016384 = 0.096256$.

Step3: Calculate the expected value for part (b)

The expected value of a binomial - distribution is $E(X)=np$. Substituting $n = 7$ and $p = 0.4$, we get $E(X)=7\times0.4 = 2.8$.

Step4: Calculate the standard deviation for part (c)

The standard deviation of a binomial - distribution is $\sigma=\sqrt{np(1 - p)}$. Substituting $n = 7$, $p = 0.4$, and $1 - p = 0.6$, we get $\sigma=\sqrt{7\times0.4\times0.6}=\sqrt{1.68}\approx1.2961$.

Answer:

a) $0.096256$ b) $2.8$ c) $\approx1.2961$