A group of men and women are asked if they prefer summer or winter the resulting two way frequency table shown.
summer winter
women 85 32
men 70 51
use the data in the table to find each probability
a. P(women)
b.P(women or summer)
c.P(woman and summer)
d.P(woman|summer)
(a) 117 women and 121 men
so P(woman) = 117/238
(b) 117 women + 70 summer men
so P = 187/238
(c) 85/238
(d) 155 summer, so P = 85/155
a. P(women) = (85+32) / (85+32+70+51) = 117 / 238
b. P(women or summer) = (85+32+70) / (85+32+70+51) = 187 / 238
c. P(woman and summer) = 85 / (85+32+70+51) = 85 / 238
d. P(woman|summer) = P(woman and summer) / P(summer)
P(woman|summer) = (85 / 238) / (85+70) / (85+32+70+51)
P(woman|summer) = (85 / 238) / (155 / 238)
P(woman|summer) = 85 / 155
To find the probabilities, we can use the formula:
Probability = Number of favorable outcomes / Total number of outcomes
a. P(women):
To find the probability of women, we need to add up the number of women who prefer summer and winter: 85 + 32 = 117. The total number of people is the sum of all four values in the table: 85 + 32 + 70 + 51 = 238.
P(women) = Number of women / Total number of people = 117 / 238 ≈ 0.4916
b. P(women or summer):
To find the probability of women or summer, we need to add up the number of women and the number of people who prefer summer: 85 + 70 = 155.
P(women or summer) = Number of women or summer / Total number of people = 155 / 238 ≈ 0.6513
c. P(woman and summer):
To find the probability of woman and summer, we directly use the value in the table: 85.
P(woman and summer) = Number of women and summer / Total number of people = 85 / 238 ≈ 0.3571
d. P(woman|summer):
To find the probability of a woman given that it is summer, we divide the number of women who prefer summer by the total number of people who prefer summer: 85.
P(woman|summer) = Number of women who prefer summer / Total number of people who prefer summer = 85 / (85 + 70) = 85 / 155 ≈ 0.5484
To find the probabilities based on the given two-way frequency table, we can use the following formulas:
a. P(women) is calculated by dividing the number of women by the total number of people in the group.
P(women) = Number of women / Total number of people
In this case, the number of women is 85 (from the table), and the total number of people is the sum of both women and men (85 + 32 + 70 + 51). Let's calculate P(women):
P(women) = 85 / (85 + 32 + 70 + 51) = 85 / 238
b. P(women or summer) is calculated by adding the number of women who prefer summer to the number of men who prefer summer, and dividing it by the total number of people.
P(women or summer) = (Number of women who prefer summer + Number of men who prefer summer) / Total number of people
From the table, the number of women who prefer summer is 85, and the number of men who prefer summer is 70. So, P(women or summer) can be calculated as:
P(women or summer) = (85 + 70) / (85 + 32 + 70 + 51)
c. P(woman and summer) refers to the probability of a person being both a woman and preferring summer. We can calculate this by dividing the number of women who prefer summer by the total number of people.
P(woman and summer) = Number of women who prefer summer / Total number of people
Using the table, the number of women who prefer summer is 85. Therefore:
P(woman and summer) = 85 / (85 + 32 + 70 + 51)
d. P(woman|summer) represents the probability of a person being a woman, given that they prefer summer. This is calculated by dividing the number of women who prefer summer by the total number of people who prefer summer.
P(woman|summer) = Number of women who prefer summer / Number of people who prefer summer
Using the table, the number of women who prefer summer is 85, and the total number of people who prefer summer is the sum of women and men who prefer summer (85 + 70). So, we can calculate P(woman|summer) as follows:
P(woman|summer) = 85 / (85 + 70)