a Area models are used to represent the probabilities that two independent events occur. The situation is represented by a square with an area of 1.
B
b Use the area model from Part A.
C
c Use the area model from Part A.
D
d Use the probabilities from Parts B and C.
E
e Are the values the same? What can we conclude?
A
aExample Solution:
B
b 81/361
C
c 9/19
D
d 9/19
E
e See solution.
Practice makes perfect
a We are asked make an area model for two spins of the roulette wheel. Remember, area models are used to represent the probability that two independent events occur by using a square with an area of 1. Let's think about possible outcomes. In each spin, the ball will land on one of three possible colors.
red ( R)
black (B)
green ( G)
In our model, we will write the results of the first spin horizontally and the result of the second spin vertically.
For each spin, the probability of landing on red is the number of red pockets, 18, divided by the total number of pockets. We can add the numbers of red, black, and green pockets to find the total number of possible outcomes.
We can find the probabilities of landing on black and green using a similar process.
P(Black) &= 2/38 or 1/19 [1em]
P( Green) &= 18/38 or 9/19
Let's include these probabilities in our area model.
Finally, let's complete the model by finding the area of each part. The area of a rectangle is the product of its length and width.
Our area model is now complete!
b We want to find the probability that the ball will land on red twice in a row. Let's use our area model.
The area of the red-red rectangle is 81361, which means that the probability that the ball will land on red twice in a row is 81361.
c We want to find the probability that the ball will land on red on the second spin, regardless of the first spin. This probability is the sum of the probabilities that the ball landed on red, black, or green on the first spin and then on red on the second spin. These probabilities are represented by the areas of the rectangles in the first column of the area model.
The probability that the ball will land on red on the second spin is 919.
d We are asked to calculate the conditional probability that the first spin was red given that the second spin is red. Recall that this probability is the probability that both spins are red divided by the probability that the second spin is red.
P( Red| Red)= P(Both Red)/P(Second Red)
These are the probabilities that we found in Parts B and C! Let's substitute the values into the expression and simplify the quotient.
The conditional probability that the first spin was red given that the second spin is red is 919.
e Let's take a look at our answers from Parts C and D to see how they compare to the probability of the ball landing on red for any single spin.
P(Second Red) &= 9/19 [1em] P( Red| Red) &= 9/19 [1em]
P( Red) &= 9/19
The probabilities are all the same! This tells us that that the probability of the ball landing on red is always the same and does not depend on the where it landed before or the number of times we have spun the roulette wheel.
