Discrete Probability Distributions Worksheet

Discrete Probability Distributions Worksheet 1. You flip four coins. Let X, the random variable, be the number of heads on all four coins. a. List the sample space for the experiment. Ans: S= {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT } b. What are the possible values for x? Ans: Possible values of X = { 0, 1, 2, 3, 4 } c. Is the random variable, x, continuous or discrete? Ans: Discrete d. Construct a probability distribution for this experiment. X 0 1 2 3 4 P(X) 1/16 4/16 6/16 4/16 1/16 2. Determine if the following are probability distributions (if no, state why). a. X P(X) b. X P(X) c. X P(X) 3 6 9 12 15 4/10 2/10 1/10 1/10 1/10 1 2 3 4 5 3/15 1/15 1/15 4/15 6/15 20 30 40 50 1.1 0.2 0.9 0.3 No. The total of the p(x) is less than 1 Yes, The total of p(x) is equals to 1 No. The total of p(x) is over 1 3. Construct a probability distribution for the data and draw a histogram for the following: a. The probabilities that a patient will have 0,1 ,2 , or 3 medical tests performed on entering a hospital are 6/15, 5/15, 3/15, and 1/15 respectively. x p(x) 0 6/15 1 5/15 2 3/15 3 1/15 (6/15) (5/15) (3/15) (1/15) b. A die is loaded in such a way that the probabilities of getting 1, 2, 3, 4, 5, and 6 are 1/2, 1/6, 1/12, 1/12, 1/12, and 1/12 respectively. x p(x) 1 ½ 2 1/6 3 1/12 4 1/12 (1/2) (1/6) (1/12) (1/12) (1/12) (1/12) 5 1/12 6 1/12 c. A box contains 3 $1 bills, 2 $5 bills, 1 $10 bill, and 1 $20 bill. x p(x) 1 3/7 5 2/7 10 1/7 20 1/7 d. A family has three children. Let X represent the number of boys. x p(x) 0 1/8 1 3/8 2 3/8 S= {(BBB),(BBG),(BGB),(BGG),(GBB),(GBG),(GGB),(GGG)} (3/8) (1/8) (3/8) (1/8) 3 1/8 Mean, Variance, and Expectation 1. From past experience, a company has found that in carton of transistors, 92% contain no defective transistors, 3% contain one defective transistor, 3% contain two defective transistors, and 2% contain three defective transistors. a. Construct a probability distribution below. b. Calculate the mean, variance, and standard deviation for the defective transistors. a. x 0 1 2 3 b. p(x) 0.92 0.03 0.03 0.02 x(px) 0 0.03 0.06 0.06 0.15 x-u -0.15 0.85 1.85 2.85 (x-u)² 0.0225 0.7225 3.4225 8.1225 ((x-u)²) (p(x)) 0.0207 0.021675 0.102675 0.16245 0.3075 0.15 0.3075 0.55 2. The number of suits sold per day at Suit World is shown in the probability distribution below. X 19 20 21 22 23 P(X) 0.2 0.2 0.3 0.2 0.1 a. Find the mean, variance, and standard deviation of the distribution. Ans: x p(x) x(px) x-u (x-u)² ((x-u)²) (p(x)) 19 0.2 3.8 -1.8 3.24 0.648 20 0.2 4 -0.8 0.64 0.128 21 0.3 6.3 0.2 0.04 0.012 22 23 0.2 0.1 4.4 2.3 20.8 1.2 2.2 1.44 4.84 0.288 0.484 1.56 0.8 1.56 1.5 b. If the manager of Suit World wants to make sure that he has enough suits for the next five days, how many should he buy to stock the store? Ans: 115 suits 3. The Bank of America VP feels that each savings account customer has, on average, three credit cards. The following distribution represents the number of credit cards people own. X 0 1 2 3 4 P(X) 0.18 0.44 0.27 0.08 0.03 a. Find the mean, variance, and standard deviation. x p(x) x(px) x-u (x-u)² ((x-u)²) (p(x)) 0 0.18 0 -1.34 1.7956 0.323208 1 0.44 0.44 -0.34 0.1156 0.050864 2 0.27 0.54 0.66 0.4356 0.117612 3 4 0.08 0.03 0.24 0.12 1.34 1.66 2.66 2.7556 7.0756 0.220448 0.212268 0.9244 1.34 0.94 0.96 b. Is the VP correct? Ans: No 4. Flower World determines the probabilities for the number of flower arrangements they deliver each day. X 6 7 8 9 10 P(X) 0.2 0.2 0.3 0.2 0.1 a. Find the mean, variance, and standard deviation. x 6 7 8 9 10 7.8 1.56 1.5 p(x) 0.2 0.2 0.3 0.2 0.1 x(px) 1.2 1.4 2.4 1.8 1 7.8 x-u -1.8 -0.8 0.2 1.2 2.2 (x-u)² 3.24 0.64 0.04 1.44 4.84 ((x-u)²) (p(x)) 0.648 0.128 0.012 0.288 0.484 1.56