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This is a binomial probability problem.

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2 parts
a

P(X>2)≈0.0120P(X > 2) \approx 0.0120

b

P(X≤2)≈0.9880P(X \le 2) \approx 0.9880

Step-by-step solution

4 steps

This is a binomial probability problem.
Let nn be the number of people selected, so n=4n=4.
Let pp be the probability that a person speaks Twi, so p=15%=0.15p = 15\% = 0.15.
Let qq be the probability that a person does not speak Twi, so q=1−p=1−0.15=0.85q = 1 - p = 1 - 0.15 = 0.85.
The probability of exactly kk successes in nn trials is given by the binomial probability formula:

P(X=k)=(nk)pkqn−kP(X=k) = \binom{n}{k} p^k q^{n-k}

where (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}.

Part a· 3 steps

more than half speak Twi

"More than half" of 4 people means X>2X > 2. So we need to find P(X=3)+P(X=4)P(X=3) + P(X=4).

  1. Calculate P(X=3)P(X=3).

    P(X=3)=(43)(0.15)3(0.85)4−3P(X=3) = \binom{4}{3} (0.15)^3 (0.85)^{4-3} P(X=3)=4!3!1!(0.15)3(0.85)1P(X=3) = \frac{4!}{3!1!} (0.15)^3 (0.85)^1 P(X=3)=4×(0.003375)×(0.85)P(X=3) = 4 \times (0.003375) \times (0.85) P(X=3)=0.011475P(X=3) = 0.011475

  2. Calculate P(X=4)P(X=4).

    P(X=4)=(44)(0.15)4(0.85)4−4P(X=4) = \binom{4}{4} (0.15)^4 (0.85)^{4-4} P(X=4)=4!4!0!(0.15)4(0.85)0P(X=4) = \frac{4!}{4!0!} (0.15)^4 (0.85)^0 P(X=4)=1×(0.00050625)×1P(X=4) = 1 \times (0.00050625) \times 1 P(X=4)=0.00050625P(X=4) = 0.00050625

  3. Sum the probabilities.

    P(X>2)=P(X=3)+P(X=4)P(X > 2) = P(X=3) + P(X=4) P(X>2)=0.011475+0.00050625P(X > 2) = 0.011475 + 0.00050625 P(X>2)=0.01198125P(X > 2) = 0.01198125

    Rounding to four decimal places:

    P(X>2)≈0.0120\boxed{P(X > 2) \approx 0.0120}

Part b

P(X≤2)≈0.9880P(X \le 2) \approx 0.9880

at most 2 speak Twi

"At most 2" means X≤2X \le 2. So we need to find P(X=0)+P(X=1)+P(X=2)P(X=0) + P(X=1) + P(X=2).
Alternatively, we can use the complement rule: P(X≤2)=1−P(X>2)P(X \le 2) = 1 - P(X > 2).

  1. Use the complement rule with the result from part (a).

    P(X≤2)=1−P(X>2)P(X \le 2) = 1 - P(X > 2) P(X≤2)=1−0.01198125P(X \le 2) = 1 - 0.01198125 P(X≤2)=0.98801875P(X \le 2) = 0.98801875

    Rounding to four decimal places:

    P(X≤2)≈0.9880\boxed{P(X \le 2) \approx 0.9880}
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