Statistics and Probability Answers

4. Find the probability of randomly selecting 4 good quarts of milk in succession from a cooler containing 25 quarts of which 5 have spoiled, by using:

(a) Theorem 3 of the product rule
(b) the concept of probability of events

3. The probability that an automobile being filled with gasoline also needs an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and the filter need changing is 0.14

(a) If a new oil filter is needed, what is the probability that the oil has to be changed
(b) If the oil has to be changed, what is the probability that a new oil filter is needed?

2. In the senior year of a high school graduating class of 100 students, 42 studied mathematics, 68 studied psychology, 54 studied history, 22 studied both mathematics and history, 25 studied both mathematics and psychology, 7 studied history but neither mathematics nor psychology, 10 studied all three subjects, and 8 did not take any of the three. Randomly select a student from the class and find the probabilities of the following events:

(a) A person enrolled in history takes all three subjects
(b) A person enrolled in history but not in mathematics and psychology

1. In an experiment to study the relationship of hypertension and smoking habits, the following data below are collected for 180 individuals, where H and NH in the table stands for Hypertension and Non-hypertension, respectively. If one of these individuals is selected at random, find the probability that the person is:

(a) not experiencing hypertension, given that the person is a heavy smoker
(b) a nonsmoker, given that the person is not experiencing hypertension
(c) either a moderate or heavy smoker, given that the person is not experiencing hypertension

In one month, a student recorded the lengths, to the nearest minute,
of each of the lectures she attended. The table below shows her data
and the calculations she made before drawing a histogram to illustrate
these data.
Calculate
Length of lecture (minutes) 50 ≤ x < 53 53 ≤ x < 55 55 ≤ x < 59 59 ≤ x < 67
Number of lectures a b 30 c
Frequency density 5 13 7.5 1.5
(i) The value of a, of b and of c.
(ii) The number of lectures attended in a month.
(iii) Using the assumed mean of A = 57.4, calculate the mean length
of lectures.
(iv) Calculate the Median and the Variance of the distribution.

RATSA is planning to enforce speed limits along Great North road by
using speed cameras at 4 different locations, L1, L2, L3 and L4. The
speed camera at L1 is operated 50% of the time, the speed camera at
L2 is operated 30% of the time, the speed camera at L3 is operated 20%
of the time and the speed camera at L4 is operated 40% of the time.
A person who is speeding from for work has probabilities 0.2, 0.1, 0.5
and 0.2 respectively, of passing through these locations.
(i) What is the probability that the person will receive a speed ticket?
(ii) If the person received the speed ticket, what is the probability
that it was at L3 where he violated speed limit rules?

The number of goals Alice scores in a soccer game is a Poisson(2.2) random variable. Her coach awards her 2k candies if she scores k goals in a game. For example, if she scores 1 goal she gets 2 candies, if she scores 2 goals she gets 4 candies, if she scores 3 goals she gets 8 candies, and so on. If Alice does not score any goals, she does not get any candies. What is the expected number of candies Alice gets in a game?

2. Two distinct six sided die are rolled. The sum of the pips facing up is recorded
(a) Use the multiplication principle to determine the number of outcomes in this experiment .
(b) Write the sample space below
(c) List the events from the Sample Space
(d) Find the probability of . . . (Please leave your solutions as unsimplified fractions)
i. sum is 4
ii. sum is 11
iii. sum ≥ 5
iv. sum ≤ 10
v. sum ≥ 5 or sum ≤ 10
vi. sum ≥ 5 and sum ≤10
vii. sum = 5 or sum = 10
viii. sum = 5 and sum = 10

1. A fair six sided die is rolled followed by a fair coin flip. Please find the following :
(a) Use the multiplication principle to determine the number of outcomes in this experiment (2pts).
(b) Write the sample space below (4pts)
(c) Find the probability of (please leave your solutions as unsimplified fractions) (1pt each)
i. the coin landing tails.
ii. the die rolled a 2
iii. a two is rolled or the coin lands tails
iv. a two is rolled and the coin lands tails.