Question
Error Analysis The sample space of picking a 2 character password using the letters Upper G, Upper R, Upper Y, Upper B, Upper O, Upper P, where double letters are not allowed is shown. Your teacher asks the class to find the probability of choosing a password with no Gs. Carlos incorrectly says that the probability is three halves
. Find the correct probability then decide which error Carlos might have made.
.
.
.
Question content area top right
Part 1
Possible Combinations
(Upper G,Upper R)
(Upper R,Upper Y)
(Upper Y,Upper B)
(Upper B,Upper O)
(Upper O,Upper P)
(Upper P,Upper G)
(Upper G,Upper Y)
(Upper R,Upper B)
(Upper Y,Upper O)
(Upper B,Upper P)
(Upper O,Upper G)
(Upper P,Upper R)
(Upper G,Upper B)
(Upper R,Upper O)
(Upper Y,Upper P)
(Upper B,Upper G)
(Upper O,Upper R)
(Upper P,Upper Y)
(Upper G,Upper O)
(Upper R,Upper P)
(Upper Y,Upper G)
(Upper B,Upper R)
(Upper O,Upper Y)
(Upper P,Upper B)
(Upper G,Upper P)
(Upper R,Upper G)
(Upper Y,Upper R)
(Upper B,Upper Y)
(Upper O,Upper B)
(Upper P,Upper O)
Question content area bottom
Part 1
The probability of choosing a password with no Gs is
two thirds
.
font size increased by 19
(Simplify your answer.)
Part 2
Which error might have Carlos made?
A.
Carlos divided the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.
B.
Carlos multiplied the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.
. Find the correct probability then decide which error Carlos might have made.
.
.
.
Question content area top right
Part 1
Possible Combinations
(Upper G,Upper R)
(Upper R,Upper Y)
(Upper Y,Upper B)
(Upper B,Upper O)
(Upper O,Upper P)
(Upper P,Upper G)
(Upper G,Upper Y)
(Upper R,Upper B)
(Upper Y,Upper O)
(Upper B,Upper P)
(Upper O,Upper G)
(Upper P,Upper R)
(Upper G,Upper B)
(Upper R,Upper O)
(Upper Y,Upper P)
(Upper B,Upper G)
(Upper O,Upper R)
(Upper P,Upper Y)
(Upper G,Upper O)
(Upper R,Upper P)
(Upper Y,Upper G)
(Upper B,Upper R)
(Upper O,Upper Y)
(Upper P,Upper B)
(Upper G,Upper P)
(Upper R,Upper G)
(Upper Y,Upper R)
(Upper B,Upper Y)
(Upper O,Upper B)
(Upper P,Upper O)
Question content area bottom
Part 1
The probability of choosing a password with no Gs is
two thirds
.
font size increased by 19
(Simplify your answer.)
Part 2
Which error might have Carlos made?
A.
Carlos divided the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.
B.
Carlos multiplied the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.
Answers
Answered by
GPT-4o mini
To find the correct probability of choosing a password with no Gs, we first determine the total number of possible combinations of 2-character passwords using the letters G, R, Y, B, O, and P where no letter can be repeated.
### Step 1: Count the Total Possible Passwords
We have 6 unique letters: G, R, Y, B, O, P. For a 2-character password where no double letters are allowed, we can choose:
- The first character: 6 options (G, R, Y, B, O, P)
- The second character: 5 remaining options (since we cannot use the letter chosen as the first character)
So, the total number of possible passwords is:
\[
6 \text{ (first choice)} \times 5 \text{ (second choice)} = 30 \text{ possible passwords}
\]
### Step 2: Count the Passwords with No Gs
Next, we find the number of favorable outcomes for passwords that do not contain the letter G. The letters available are R, Y, B, O, and P (5 letters total):
- The first character: 5 options (R, Y, B, O, P)
- The second character: 4 remaining options
Thus, the number of valid passwords without G is:
\[
5 \text{ (first choice)} \times 4 \text{ (second choice)} = 20 \text{ favorable passwords}
\]
### Step 3: Calculate the Probability
The probability \( P \) of choosing a password with no Gs is given by the ratio of favorable outcomes to possible outcomes:
\[
P(\text{no Gs}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{20}{30} = \frac{2}{3}
\]
### Summary of Probability
Thus, the correct probability of choosing a password with no Gs is \( \frac{2}{3} \).
### Part 2: Analyzing Carlos's Error
Carlos claimed the probability is three halves, which is greater than 1, which is impossible. This indicates that he must have made a significant error in his calculations.
- **Option A** states that Carlos divided the number of possible outcomes by the number of favorable outcomes instead of the reverse. This would lead to an incorrect probability larger than 1.
- **Option B** suggests that he multiplied the numbers instead of dividing, which would also result in a nonsensical value that wouldn't be established as a probability.
Considering the options and the error Carlos made, it seems most likely that:
### Correct Choice
**A. Carlos divided the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.**
### Step 1: Count the Total Possible Passwords
We have 6 unique letters: G, R, Y, B, O, P. For a 2-character password where no double letters are allowed, we can choose:
- The first character: 6 options (G, R, Y, B, O, P)
- The second character: 5 remaining options (since we cannot use the letter chosen as the first character)
So, the total number of possible passwords is:
\[
6 \text{ (first choice)} \times 5 \text{ (second choice)} = 30 \text{ possible passwords}
\]
### Step 2: Count the Passwords with No Gs
Next, we find the number of favorable outcomes for passwords that do not contain the letter G. The letters available are R, Y, B, O, and P (5 letters total):
- The first character: 5 options (R, Y, B, O, P)
- The second character: 4 remaining options
Thus, the number of valid passwords without G is:
\[
5 \text{ (first choice)} \times 4 \text{ (second choice)} = 20 \text{ favorable passwords}
\]
### Step 3: Calculate the Probability
The probability \( P \) of choosing a password with no Gs is given by the ratio of favorable outcomes to possible outcomes:
\[
P(\text{no Gs}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{20}{30} = \frac{2}{3}
\]
### Summary of Probability
Thus, the correct probability of choosing a password with no Gs is \( \frac{2}{3} \).
### Part 2: Analyzing Carlos's Error
Carlos claimed the probability is three halves, which is greater than 1, which is impossible. This indicates that he must have made a significant error in his calculations.
- **Option A** states that Carlos divided the number of possible outcomes by the number of favorable outcomes instead of the reverse. This would lead to an incorrect probability larger than 1.
- **Option B** suggests that he multiplied the numbers instead of dividing, which would also result in a nonsensical value that wouldn't be established as a probability.
Considering the options and the error Carlos made, it seems most likely that:
### Correct Choice
**A. Carlos divided the number of possible outcomes by the number of favorable outcomes. He should have divided the number of favorable outcomes by the number of possible outcomes.**