Question

hw 9 - binomial distributions
score: 3.28/10 answered: 4/10
question 7
about 20% of seniors age 75 and older started using delivery services after the covid - 19 pandemic. if randomly select 10 seniors age 75 and older in the united states, assuming a binomial distribution, find the following probabilities.
round your answers to 4 decimal places.
(a) what is the probability that exactly 3 of them started using delivery services after the covid - 19 pandemic?
(b) what is the probability at least 3 of them started using delivery services after the covid - 19 pandemic?

Explanation:

Part (a)

Step 1: Identify Binomial Parameters

We have a binomial distribution with \( n = 10 \) (number of trials), \( p = 0.2 \) (probability of success), and we want \( k = 3 \) (number of successes). The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \), where \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \).

Step 2: Calculate the Combination

First, calculate \( \binom{10}{3}=\frac{10!}{3!(10 - 3)!}=\frac{10!}{3!7!}=\frac{10\times9\times8}{3\times2\times1}=120 \).

Step 3: Calculate the Probability

Now, substitute into the formula: \( P(X = 3)=\binom{10}{3}(0.2)^3(0.8)^{7} \).
\( (0.2)^3 = 0.008 \), \( (0.8)^7\approx0.2097152 \).
Multiply them together: \( 120\times0.008\times0.2097152 = 120\times0.0016777216 = 0.201326592 \).

Step 4: Round to 4 Decimals

Rounding \( 0.201326592 \) to 4 decimal places gives \( 0.2013 \).

Step 1: Understand "At Least 3"

"At least 3" means \( P(X\geq3)=1 - P(X < 3)=1 - [P(X = 0)+P(X = 1)+P(X = 2)] \).

Step 2: Calculate \( P(X = 0) \)

Using the binomial formula with \( k = 0 \):
\( P(X = 0)=\binom{10}{0}(0.2)^0(0.8)^{10} \).
\( \binom{10}{0}=1 \), \( (0.2)^0 = 1 \), \( (0.8)^{10}\approx0.1073741824 \).
So \( P(X = 0)=1\times1\times0.1073741824 = 0.1073741824 \).

Step 3: Calculate \( P(X = 1) \)

With \( k = 1 \):
\( \binom{10}{1}=\frac{10!}{1!9!}=10 \), \( (0.2)^1 = 0.2 \), \( (0.8)^9\approx0.134217728 \).
\( P(X = 1)=10\times0.2\times0.134217728 = 10\times0.0268435456 = 0.268435456 \).

Step 4: Calculate \( P(X = 2) \)

With \( k = 2 \):
\( \binom{10}{2}=\frac{10!}{2!8!}=\frac{10\times9}{2\times1}=45 \), \( (0.2)^2 = 0.04 \), \( (0.8)^8\approx0.16777216 \).
\( P(X = 2)=45\times0.04\times0.16777216 = 45\times0.0067108864 = 0.301989888 \).

Step 5: Sum the Probabilities for \( X < 3 \)

\( P(X < 3)=P(X = 0)+P(X = 1)+P(X = 2)=0.1073741824 + 0.268435456 + 0.301989888 = 0.6777995264 \).

Step 6: Calculate \( P(X\geq3) \)

\( P(X\geq3)=1 - 0.6777995264 = 0.3222004736 \).

Step 7: Round to 4 Decimals

Rounding \( 0.3222004736 \) to 4 decimal places gives \( 0.3222 \).

Answer:

(a):
\( 0.2013 \)

Part (b)