Given that X is a binomial distribution with n=8 and p=0.6. The probability mass function is P(X=k) = nk p^k…

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Given that X is a binomial distribution with n=8 and p=0.6. The probability mass function is P(X=k) = nk p^k (1-p)ⁿ−k.

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Given that X is a binomial distribution with n=8 and p=0.6. The probability mass function is P(X=k) = nk p^k (1−p)^(n−k). Here, n=8, p=0.6, and 1−p = 0.4. a) Find P(X=5) Step 1: Apply the binomial probability formula. P(X=5) = 85 (0.6)⁵ (0.4)⁸⁻⁵; P(X=5) = 85 (0.6)⁵ (0.4)³ Step 2: Calculate the binomial coefficient. 85 = (8!)/(5!(8−5)!) = (8!)/(5!3!) = (8 × 7 × 6)/(3 × 2 × 1) = 56 Step 3: Calculate the powers of p and (1−p). (0.6)⁵ = 0.07776; (0.4)³ = 0.064 Step 4: Multiply the values. P(X=5) = 56 × 0.07776 × 0.064 = 0.27869184 Step 5: Round to 4 decimal places. P(X=5) ≈ 0.2787 The final answer is 0.2787 b) Find P(X > 6) Step 1: Express P(X > 6) as a sum of probabilities. P(X > 6) = P(X=7) + P(X=8); Step 2: Calculate P(X=7). P(X=7) = 87 (0.6)⁷ (0.4)⁸⁻⁷ = 87 (0.6)⁷ (0.4)¹; 87 = 8; (0.6)⁷ = 0.0279936; P(X=7) = 8 × 0.0279936 × 0.4 = 0.08957952; Step 3: Calculate P(X=8). P(X=8) = 88 (0.6)⁸ (0.4)⁸⁻⁸ = 88 (0.6)⁸ (0.4)⁰; 88 = 1; (0.6)⁸ = 0.01679616; P(X=8) = 1 × 0.01679616 × 1 = 0.01679616 Step 4: Sum the probabilities. P(X > 6) = 0.08957952 + 0.01679616 = 0.10637568 Step 5: Round to 4 decimal places. P(X > 6) ≈ 0.1064 The final answer is 0.1064 c) Find P(2 < X < 6) Step 1: Express P(2 < X < 6) as a sum of probabilities. P(2 < X < 6) = P(X=3) + P(X=4) + P(X=5); Step 2: Calculate P(X=3). P(X=3) = 83 (0.6)³ (0.4)⁸⁻³ = 83 (0.6)³ (0.4)⁵; 83 = (8!)/(3!5!) = (8 × 7 × 6)/(3 × 2 × 1) = 56; (0.6)³ = 0.216; (0.4)⁵ = 0.01024; P(X=3) = 56 × 0.216 × 0.01024 = 0.12386304; Step 3: Calculate P(X=4). P(X=4) = 84 (0.6)⁴ (0.4)⁸⁻⁴ = 84 (0.6)⁴ (0.4)⁴; 84 = (8!)/(4!4!) = (8 × 7 × 6 × 5)/(4 × 3 × 2 × 1) = 70; (0.6)⁴ = 0.1296; (0.4)⁴ = 0.0256; P(X=4) = 70 × 0.1296 × 0.0256 = 0.2322432; Step 4: Use P(X=5) from part a). P(X=5) = 0.27869184 Step 5: Sum the probabilities. P(2 < X < 6) = 0.12386304 + 0.2322432 + 0.27869184 = 0.63479808 Step 6: Round to 4 decimal places. P(2 < X < 6) ≈ 0.6348 The final answer is 0.6348

Given that X is a binomial distribution with n=8 and p=0.6. The probability mass function is P(X=k) = nk p^k (1-p)ⁿ-k.

Answers

Step-by-step solution

Part a· 5 steps

Apply the binomial probability formula.

Calculate the binomial coefficient.

Calculate the powers of $p$ and $(1-p)$ .

Multiply the values.

Round to 4 decimal places.

Part b· 5 steps
$\text{0.1064}$

Express $P(X > 6)$ as a sum of probabilities.

Calculate $P(X=7)$ .

Calculate $P(X=8)$ .

Sum the probabilities.

Round to 4 decimal places.

Part c· 6 steps
$\text{0.6348}$

Express $P(2 < X < 6)$ as a sum of probabilities.

Calculate $P(X=3)$ .

Calculate $P(X=4)$ .

Use $P(X=5)$ from part a).

Sum the probabilities.

Round to 4 decimal places.

Try one yourself

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Given that X is a binomial distribution with n=8 and p=0.6. The probability mass function is P(X=k) = nk p^k (1-p)ⁿ-k.

Answers

Step-by-step solution

Part a· 5 steps

Apply the binomial probability formula.

Calculate the binomial coefficient.

Calculate the powers of ppp and (1−p)(1-p)(1−p).

Multiply the values.

Round to 4 decimal places.

Part b· 5 steps0.1064\text{0.1064}0.1064

Express P(X>6)P(X > 6)P(X>6) as a sum of probabilities.

Calculate P(X=7)P(X=7)P(X=7).

Calculate P(X=8)P(X=8)P(X=8).

Sum the probabilities.

Round to 4 decimal places.

Part c· 6 steps0.6348\text{0.6348}0.6348

Express P(2<X<6)P(2 < X < 6)P(2<X<6) as a sum of probabilities.

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

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

Use P(X=5)P(X=5)P(X=5) from part a).

Sum the probabilities.

Round to 4 decimal places.

Try one yourself

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Calculate the powers of $p$ and $(1-p)$ .

Part b· 5 steps
$\text{0.1064}$

Express $P(X > 6)$ as a sum of probabilities.

Calculate $P(X=7)$ .

Calculate $P(X=8)$ .

Part c· 6 steps
$\text{0.6348}$

Express $P(2 < X < 6)$ as a sum of probabilities.

Calculate $P(X=3)$ .

Calculate $P(X=4)$ .

Use $P(X=5)$ from part a).