A survey is taken of 400 people asking what type of cell phone they own and what cell phone carrier they use. The results are shown in the table below.
Carrier
Phone Type Verizon T-Mobile AT&T
Total
Android 40 110 65 215
iPhone 50 65 70 185
Total 90 175 135 400
Find the following probabilities from the given table. Probabilities should be written as decimals rounded to 3 decimal places.
P(Verizon) =
P(iPhone) =
P(Verizon|iPhone) =
P(iPhone|Verizon) =
Are the events “iPhone” and “Verizon” independent?
P(Verizon) = (Number of people with Verizon)/(Total number of people) = 90/400 = 0.225
P(iPhone) = (Number of people with iPhone)/(Total number of people) = 185/400 = 0.463
P(Verizon|iPhone) = (Number of people with iPhone and Verizon)/(Number of people with iPhone) = 50/185 = 0.270
P(iPhone|Verizon) = (Number of people with iPhone and Verizon)/(Number of people with Verizon) = 50/90 = 0.556
The events "iPhone" and "Verizon" are not independent because P(Verizon|iPhone) is not equal to P(Verizon) and P(iPhone|Verizon) is not equal to P(iPhone).
To find the probabilities, we can use the information provided in the table.
P(Verizon) = Number of people with Verizon / Total number of people
P(Verizon) = 90 / 400 = 0.225
P(iPhone) = Number of people with iPhone / Total number of people
P(iPhone) = 185 / 400 = 0.463
To find P(Verizon|iPhone):
P(Verizon|iPhone) = Number of people with Verizon and iPhone / Number of people with iPhone
P(Verizon|iPhone) = 50 / 185 = 0.270
To find P(iPhone|Verizon):
P(iPhone|Verizon) = Number of people with iPhone and Verizon / Number of people with Verizon
P(iPhone|Verizon) = 50 / 90 = 0.556
To determine if the events "iPhone" and "Verizon" are independent, we need to compare P(Verizon|iPhone) with P(Verizon), and P(iPhone|Verizon) with P(iPhone):
If P(Verizon|iPhone) = P(Verizon), then the events are independent.
If P(iPhone|Verizon) = P(iPhone), then the events are independent.
In this case, P(Verizon|iPhone) = 0.270 and P(Verizon) = 0.225, so the events "iPhone" and "Verizon" are not independent.
Similarly, P(iPhone|Verizon) = 0.556 and P(iPhone) = 0.463, so the events "iPhone" and "Verizon" are not independent.
To find the probabilities, we need to use the information given in the table. Let's solve the problem step by step.
1. P(Verizon):
This can be found by dividing the number of Verizon users (90) by the total number of people surveyed (400):
P(Verizon) = 90/400 = 0.225
2. P(iPhone):
This can be found by dividing the number of iPhone users (185) by the total number of people surveyed (400):
P(iPhone) = 185/400 = 0.463
3. P(Verizon|iPhone):
This represents the probability of having Verizon given that the person has an iPhone.
To find this probability, we need to divide the number of people who have Verizon and iPhone (50) by the number of iPhone users (185):
P(Verizon|iPhone) = 50/185 = 0.270
4. P(iPhone|Verizon):
This represents the probability of having an iPhone given that the person is on Verizon.
To find this probability, we need to divide the number of people who have Verizon and iPhone (50) by the number of Verizon users (90):
P(iPhone|Verizon) = 50/90 = 0.556
5. Are the events "iPhone" and "Verizon" independent?
For two events to be considered independent, the occurrence of one event should not affect the probability of the other event. To determine if the events "iPhone" and "Verizon" are independent, we can compare P(Verizon|iPhone) and P(Verizon).
If P(Verizon|iPhone) = P(Verizon), then the events are independent.
In our case, P(Verizon|iPhone) = 0.270 and P(Verizon) = 0.225. Since these probabilities are not equal, the events "iPhone" and "Verizon" are not independent.