University:
Jackson College
Course:
MAT-010 | Basic Mathematics
Academic year:

2024
Views:

155

Pages:

3
Author:

Bennett Wheeler

Question 1 How many 5-card poker hands contain: (a) four "10"s? (b) any "four of a kind"? (c) aces full of kings (3 A + 2 K)? (d) 2 red Aces, 2 black Kings, and 1 red Queen? (e) 2 Aces, 2 Kings, and 1 Queen? Answer 1 (a) Four tens (always) + any card: There are 52 − 4 = 48 52−4=48 possible hands with four 10s. (b) For four of a kind: There are 4 combinations of AAA and 6 combinations of KK, so 4 × 6 = 24 4×6=24. (c) Aces full of kings: 24 combinations using 13 choices for "3 of a kind" and any remaining 12 for the "pair," totaling 24 × 13 × 12 = 3744 24×13×12=3744. (d) One red A, one black K, and two single red Q's, so 1 × 1 × 2 = 2 1×1×2=2. (e) 6 combinations of pairs for Aces, Kings, and Queens, totaling 6 × 6 × 4 = 144 6×6×4=144. Question 2 An automatic machine in a manufacturing process operates correctly if the lengths of an important subcomponent are normally distributed with a mean of 116 cm and a standard deviation of 4.8 cm. A. Find the probability that one subcomponent is longer than 118 cm. B. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 118 cm. C. Find the probability that all 4 have lengths exceeding 118 cm. Answer 1 Given data: 𝑋 ∼ 𝑁 ( 116 , 4. 8 2 ) X∼N(116,4.8 2 ) A. 𝑃 ( 𝑋 > 118 ) P(X>118) is calculated as 𝑃 ( 𝑍 > 118 − 116 4.8 ) = 0.338 P(Z> 4.8 118−116 )=0.338. B. For sample size 𝑛 = 4 n=4: 𝑃 ( 𝑋 ˉ > 118 ) = 0.202 P( X ˉ >118)=0.202. C. 𝑃 ( all 4 exceed 118 cm ) = 0.33 8 4 = 0.0131 P(all 4 exceed 118 cm)=0.338 4 =0.0131. Question 3 Find the mean and standard deviation of a standard normal distribution. Answer 1 For a standard normal distribution, 𝜇 = 0 μ=0 and 𝜎 = 1 σ=1. Question 4 Find 𝑃 ( 𝑍 > 2.5 ) P(Z>2.5) for a standard normal distribution. Answer 1 Using symmetry, 𝑃 ( 𝑍 > 2.5 ) = 1 − 0.4938 = 0.0062 P(Z>2.5)=1−0.4938=0.0062. Question 5 Explain some characteristics of a normal distribution. Answer 1 Symmetrical around the mean. The area under the curve equals 1. Bell-shaped