Question 1174702: Q.No.4. (a) Sixty percent of all buyers of new cars choose automatic transmissions. For a group of five buyers of new cars, calculate the mean and standard deviation for the number of buyers choosing automatic transmissions. Also calculate the complete distribution.
(b) What is the probability that a poker hand of 5 cards contain exactly 2 king.
Q: If x is a passion random variable with parameter value 1.5, find p(x=1) and P (X = 7).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down each part of this problem.
**(a) Automatic Transmissions**
* **Binomial Distribution:** This situation follows a binomial distribution because:
* There are a fixed number of trials (n = 5 buyers).
* Each trial has only two outcomes (automatic or not).
* The probability of success (automatic) is constant (p = 0.6).
* The trials are independent.
* **Mean (μ):**
* μ = np = 5 * 0.6 = 3
* **Standard Deviation (σ):**
* σ = √(np(1-p)) = √(5 * 0.6 * 0.4) = √1.2 ≈ 1.095
* **Complete Distribution:**
* P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
* Where (n choose k) = n! / (k! * (n-k)!)
* P(X = 0) = (5 choose 0) * (0.6)^0 * (0.4)^5 = 1 * 1 * 0.01024 = 0.01024
* P(X = 1) = (5 choose 1) * (0.6)^1 * (0.4)^4 = 5 * 0.6 * 0.0256 = 0.0768
* P(X = 2) = (5 choose 2) * (0.6)^2 * (0.4)^3 = 10 * 0.36 * 0.064 = 0.2304
* P(X = 3) = (5 choose 3) * (0.6)^3 * (0.4)^2 = 10 * 0.216 * 0.16 = 0.3456
* P(X = 4) = (5 choose 4) * (0.6)^4 * (0.4)^1 = 5 * 0.1296 * 0.4 = 0.2592
* P(X = 5) = (5 choose 5) * (0.6)^5 * (0.4)^0 = 1 * 0.07776 * 1 = 0.07776
**(b) Poker Hand with 2 Kings**
* **Total Possible Hands:** (52 choose 5) = 2,598,960
* **Ways to Choose 2 Kings:** (4 choose 2) = 6
* **Ways to Choose 3 Other Cards (not Kings):** (48 choose 3) = 17,296
* **Probability:** (6 * 17,296) / 2,598,960 ≈ 0.0399
**(c) Poisson Random Variable**
* **Poisson Distribution:** X ~ Poisson(λ), where λ = 1.5.
* **Formula:** P(X = k) = (e^(-λ) * λ^k) / k!
* **P(X = 1):**
* P(X = 1) = (e^(-1.5) * 1.5^1) / 1!
* P(X = 1) ≈ (0.2231 * 1.5) / 1 ≈ 0.3347
* **P(X = 7):**
* P(X = 7) = (e^(-1.5) * 1.5^7) / 7!
* P(X=7) = (0.2231 * 17.0859)/5040
* P(X=7) = 3.811/5040
* P(X = 7) ≈ 0.000756
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