Question 1 In $$ 2013,16.4 \% $$ of Tanzanians had access to electricity. What is the probability that for a random sample of 10 Tanzanians, 2 have access to electricity? Round to four decimal places. Answer Check

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UpStudy AI · Accepted Answer

To find the probability that exactly 2 out of 10 Tanzanians have access to electricity, we can use the binomial probability formula:

$$
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
$$

where:
- $$ n $$ is the number of trials (in this case, 10),
- $$ k $$ is the number of successes (in this case, 2),
- $$ p $$ is the probability of success (in this case, $$ 0.164 $$),
- $$ \binom{n}{k} $$ is the binomial coefficient, calculated as $$ \frac{n!}{k!(n-k)!} $$.

Given:
- $$ n = 10 $$
- $$ k = 2 $$
- $$ p = 0.164 $$

First, we calculate the binomial coefficient $$ \binom{10}{2} $$:

$$
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
$$

Next, we calculate $$ p^k $$ and $$ (1-p)^{n-k} $$:

$$
p^k = (0.164)^2 = 0.026896
$$

$$
1 - p = 1 - 0.164 = 0.836
$$

$$
(1-p)^{n-k} = (0.836)^{10-2} = (0.836)^8 \approx 0.436
$$

Now we can plug these values into the binomial probability formula:

$$
P(X = 2) = \binom{10}{2} \cdot (0.164)^2 \cdot (0.836)^8
$$

$$
P(X = 2) = 45 \cdot 0.026896 \cdot 0.436
$$

Calculating this gives:

$$
P(X = 2) \approx 45 \cdot 0.026896 \cdot 0.436 \approx 0.0005
$$

Now, let's calculate it step by step:

1. Calculate $$ 45 \cdot 0.026896 $$:
   $$
   45 \cdot 0.026896 \approx 1.20632
   $$

2. Now multiply by $$ 0.436 $$:
   $$
   1.20632 \cdot 0.436 \approx 0.525
   $$

Thus, the final probability is:

$$
P(X = 2) \approx 0.525
$$

Rounding to four decimal places, we get:

$$
\boxed{0.5250}
$$
The probability that exactly 2 out of 10 Tanzanians have access to electricity is approximately 0.5250.

Question 1 In \( 2013,16.4 \% \) of Tanzanians had access to electricity. What is the probability that for a random sample of 10 Tanzanians, 2 have access to electricity? Round to four decimal places. Answer Check

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