Solved Worksheet - Probability Addition and Multiplication | Chegg.com - Free Printable

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Problem 1: The Spinner

a. Give the sample space for two consecutive spins.
The spinner has three colors: Orange (O), Yellow (Y), and Green (G). Since we are spinning it twice, we list every possible combination of the first spin and the second spin.

* If the first is Orange: OO, OY, OG
* If the first is Yellow: YO, YY, YG
* If the first is Green: GO, GY, GG

There are $3 \times 3 = 9$ total outcomes.

b. Find the probability of observing at least 1 orange.
"At least 1 orange" means getting one orange or two oranges. Let's look at our list from part (a):
* OO (2 oranges) - Yes
* OY (1 orange) - Yes
* OG (1 orange) - Yes
* YO - No
* YY - No
* YG - No
* GO (1 orange) - Yes
* GY - No
* GG - No

There are 4 favorable outcomes out of 9 total outcomes.
Probability = $\frac{4}{9}$

c. Find the conditional probability that the first spin is orange given that there is at least 1 orange spin.
Conditional probability asks us to look *only* at the cases where the condition is met.
* Condition: "At least 1 orange." From part (b), we know there are 4 such outcomes: {OO, OY, OG, GO}. This is our new denominator.
* Event: "First spin is orange." Looking at those 4 specific outcomes, which ones start with Orange?
* OO (Starts with O)
* OY (Starts with O)
* OG (Starts with O)
* GO (Starts with G) -> No

There are 3 outcomes where the first spin is orange.
Probability = $\frac{3}{4}$

***

Problem 2: Senate Votes

First, let's find the totals for the table to make calculations easier.
* Total Democrats: $23 + 21 = 44$
* Total Republicans: $43 + 7 = 50$
* Total Independents: $2 + 4 = 6$
* Total Senators: $44 + 50 + 6 = 100$

a. The probability the voter is a Democrat and against the bill.
We look for the specific cell in the table where "Democrat" and "Against" meet.
* Number of Democrats against: 21
* Total voters: 100
Probability = $\frac{21}{100}$ or $0.21$

b. The probability the voter is Republican.
We add up all the Republicans (In Favor + Against).
* Total Republicans: $43 + 7 = 50$
* Total voters: 100
Probability = $\frac{50}{100} = \frac{1}{2}$ or $0.5$

c. The probability the voter is Republican or voted against the bill.
For "OR" probabilities, we use the formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
* P(Republican): 50 people.
* P(Against): Add the "Against" column ($21 + 7 + 4 = 32$ people).
* P(Republican AND Against): We counted the Republicans who voted against twice (once in the Republican total, once in the Against total). We must subtract them so they aren't double-counted. There are 7 Republicans who voted against.

Calculation: $50 + 32 - 7 = 75$ people.
Probability = $\frac{75}{100} = \frac{3}{4}$ or $0.75$

d. The conditional probability the voter is against the bill given that the voter is Democrat.
The phrase "given that the voter is Democrat" tells us to ignore everyone else. We only look at the Democrat row.
* Total Democrats: 44 (This is our denominator).
* Democrats Against: 21 (This is our numerator).

Probability = $\frac{21}{44}$

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Final Answer:

1. Spinner Problem
a. Sample Space: {OO, OY, OG, YO, YY, YG, GO, GY, GG}
b. Probability: $\frac{4}{9}$
c. Conditional Probability: $\frac{3}{4}$

2. Senate Vote Problem
a. Probability: $\frac{21}{100}$ (or 0.21)
b. Probability: $\frac{1}{2}$ (or 0.5)
c. Probability: $\frac{3}{4}$ (or 0.75)
d. Conditional Probability: $\frac{21}{44}$