Use the fact that the probability of a compound event is the sum of the probabilities of each individual event. Calculate each probability using the Probability Formula.
A
a
4.7 %
B
b
57.4 %
Practice makes perfect
a
Let X represent the number of uses of the GPS system by a random user. We want to find the probability that a randomly chosen user will have used the GPS more than 20 times, which is the same as finding the probability that the value of our random variable X is greater than 20.
P(X>20)= ?
Before we start calculating let's look at the given table.
Uses
Customers
0
1382
1-5
2350
6-10
2010
11-15
1863
16-20
1925
21+
470
Since X represents the number of uses of a GPS, we can rewrite the first column using our random variable X.
Value of X
Customers
X=0
1382
1≤ X≤ 5
2350
6≤ X ≤ 10
2010
11≤ X ≤ 15
1863
16≤ X ≤ 20
1925
X ≥ 21
470
To find P(X>20) we need to find the number of customers that have used the GPS system more than 20 times. Looking at the table we can tell that there is only one outcome where X>20. To find the probability, we will use the Probability Formula.
P(X>20)=Customers for whom X>20/Customers surveyed
We are given that the car dealership surveyed 10 000 of its customers with a GPS and we have found that the number of customers for whom X>20 is 470.
P(X>20)=470/10 000=0.047
The probability that a randomly chosen customer will have used the GPS system more than 20 times is equal to 0.047, which can also be written as 4.7 %.
b
This time we want to find the probability that a randomly chosen customer will have used the GPS system no more than 10 times. This is the same as finding the probability that the value of our random variable X is less than or equal to 10.
P(X ≤ 10)= ?
Let's look at the table we made in Part A again.
Value of X
Customers
X=0
1382
1≤ X≤ 5
2350
6≤ X ≤ 10
2010
11≤ X ≤ 15
1863
16≤ X ≤ 20
1925
X ≥ 21
470
We will consider the first 3 rows of our table only because the value of X is less than or equal to 10 in these rows. Our event is made up of 3 more simple events, so this is a compound event. Since the 3 events are mutually exclusive events, we can calculate P(X ≤ 10) by adding the probabilities that each event occurs.
P(X ≤ 10) &= P(X = 0) + P(1≤ X ≤ 5)& + P(6 ≤ X ≤ 10)
We will calculate each probability using the Probability Formula. Probability of each event is equal to the number of customers that are related to the given event divided by the total number of customers which is 10000.
Value of X
Customers
Probability
Simplified
X=0
1382
1382/10 000
0.1382
1≤ X ≤ 5
2350
2350/10 000
0.235
6≤ X ≤ 10
2010
2010/10 000
0.201
Now we can substitute in the values from the table into our equation for P(X≤ 10).
P(X≤ 10) &= 0.1382 + 0.235+0.201 &= 0.5742 ≈ 0.574
The probability that a randomly chosen customer will have used the GPS system no more than 10 times is equal to 0.574, which can also be written as 57.4 %.
