Question

No participant threw the same number of bullseyes on two different days. If a participant is selected at random, what is the probability that the selected participant threw 3 bullseyes on Day 1 or Day 2, given that the contestant threw 3 bullseyes on one of the three days? $$ \begin{aligned} &\text { Number of Participants by Number of Bullseyes Thrown and Day }\\\ &\begin{array}{|l|c|c|c|c|} \hline & \text { Day 1 } & \text { Day } 2 & \text { Day } 3 & \text { Total } \\\ \hline \text { 0 Bullseyes } & 2 & 3 & 4 & 9 \\ \hline \text { 1 Bullseyes } & 1 & 3 & 1 & 5 \\ \hline \text { 2 Bullseyes } & 2 & 3 & 7 & 12 \\ \hline \text { 3 Bullseyes } & 5 & 2 & 1 & 8 \\ \hline \text { 4 Bullseyes } & 3 & 2 & 0 & 5 \\ \hline \text { 5 Bullseyes } & 2 & 2 & 2 & 6 \\ \hline \text { Total } & 15 & 15 & 15 & 45 \\ \hline \end{array} \end{aligned} $$

Short answer

The probability that the selected participant threw 3 bullseyes on Day 1 or Day 2, given that the contestant threw 3 bullseyes on one of the three days is \(\frac{7}{8}\).

Step-by-step solution

To find the probability of both events A and B happening, we need to determine the number of participants who threw 3 bullseyes on Day 1 or Day 2. From the table, we can see that there are 5 participants who threw 3 bullseyes on Day 1 and 2 participants who threw 3 bullseyes on Day 2. Therefore, there are a total of 7 participants who meet the criteria for both A and B. The probability of both events happening is the ratio of these participants to the total number of participants, which is 45. So, P(A and B) = 7/45. #Step 2: Calculate P(B)#

Next, we need to calculate the probability of event B, meaning the participant threw 3 bullseyes on one of the three days. From the table, we can see that there are a total of 8 participants who threw 3 bullseyes across all three days. So, the probability of event B is P(B) = 8/45. #Step 3: Calculate P(A|B) using the conditional probability formula#

Now we can use the conditional probability formula to calculate the probability of event A given event B: $$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{7/45}{8/45} = \frac{7}{8}$$ #Step 4: Write down the final answer#

The probability that the selected participant threw 3 bullseyes on Day 1 or Day 2, given that the contestant threw 3 bullseyes on one of the three days is $$\frac{7}{8}$$.

Key concepts

Conditional Probability

Understanding conditional probability is crucial for tackling a range of problems on the SAT, especially in scenarios where the outcome of one event influences the likelihood of another. In the context of the SAT math section, conditional probability determines the probability of an event occurring (Event A), given that another event has occurred (Event B). This is denoted as `P(A|B)`.

Let's apply this to an SAT-style question. Consider the situation where we're trying to find the likelihood that a participant threw 3 bullseyes on either Day 1 or Day 2, knowing that they hit 3 bullseyes on one of the three days. In this case, Event A represents throwing 3 bullseyes on Day 1 or Day 2, and Event B is hitting 3 bullseyes on any of the three days.

The formula to calculate conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
which simplifies the process to a division of probabilities. Understanding and being able to apply this formula is imperative for success in these types of probability questions on the SAT.

Probability Formulas SAT

Probability can seem daunting, but it's all about mastering a few key formulas. The SAT math section often includes probability questions that require you to understand and apply these formulas correctly to solve problems efficiently. Some of the essential probability formulas include the basic probability of an event `P(A)` calculated as the number of favorable outcomes over the total number of possible outcomes, the probability of multiple independent events occurring together `P(A and B) = P(A) × P(B)`, and the addition rule for non-mutually exclusive events `P(A or B) = P(A) + P(B) - P(A and B)`.

In the provided exercise, we already determined `P(A and B)` and `P(B)`, then used the conditional probability formula (related to these probabilities) to solve the problem. Understanding how to manipulate these formulas based on the given information is a critical skill for SAT test-takers.

Statistics SAT Math Section

The statistics portion of the SAT math section covers a variety of topics including data interpretation, mean, median, mode, range, standard deviation, and of course, probability. Being comfortable with reading and extracting information from charts, tables, and graphs is as important as knowing the formulas.

For instance, in the example problem, the student first needed to interpret the data in the given table to extract the number of participants who hit 3 bullseyes on specified days. This step is essential and requires an understanding of statistics to accurately identify the required data before moving on to apply probability formulas.

It's also important to remember that in statistics problems on the SAT, details matter. Careful reading and understanding of what is being asked will assist in selecting the correct numbers from a table and thereby reaching the right solution. Practice with different types of data presentations is recommended to increase accuracy and speed on test day.