What is the expected number of tickets she will earn after one throw? Make sure to take the probability of hitting each field into account.
Yes.
Practice makes perfect
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.
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.
