Exercise 82 Page 289

Let's first determine the probability of hitting each field with a single dart. To identify these we have to determine how large an area each sector occupies of the circle. We see that two sectors, 2 and 3, have a 90^(∘) angle, so the remaining sector has a 180^(∘) angle. We can now calculate how much of the circle each sector occupies.

If we multiply the probability of hitting each sector by their respective number of tickets and add the products, we get the number of tickets we expect to get from one throw. Probability is calculated by dividing the number of favorable outcomes by the number of possible outcomes.

P=Number of favorable outcomes/Number of possible outcomes Let's calculate the expected number of tickets we will get with one throw.

event	P	tickets	P * tickets	expected tickets
P(2)	1/4	2	1/4( 2)	1/2
P(3)	1/4	3	1/4( 3)	3/4
P(5)	1/2	5	1/2( 5)	5/2

Now we will add the expected number of tickets we earn when throwing the dart once.

1/2+3/4+5/2

ExpandFrac

a/b=a * 2/b * 2

2/4+3/4+10/4

AddFrac

Add fractions

15/4

CalcQuot

Calculate quotient

3.75

We are expected to get 3.75 tickets when throwing the dart once. This means that after 3 throws we are expected to win 3(3.75)= 11.25 tickets. Therefore, our expectation is that she will get more than 10 tickets, which is enough.