Question
Find the probability of the given event. You must show all steps, and you may give your final answer as an exact value, a decimal to 3 places, or a percentage to the nearest tenth.
In the game of Scrabble, there are a total of 100 tiles. Of the 100 tiles, 42 tiles have the vowels A, E, I, O, and U printed on them, 56 tiles have the consonants printed on them, and 2 tiles are left blank. If tiles drawn are not replaced, what is the probability that the first tiles selected is a consonant, and the second tile is blank?
In the game of Scrabble, there are a total of 100 tiles. Of the 100 tiles, 42 tiles have the vowels A, E, I, O, and U printed on them, 56 tiles have the consonants printed on them, and 2 tiles are left blank. If tiles drawn are not replaced, what is the probability that the first tiles selected is a consonant, and the second tile is blank?
Answers
GPT-4o mini
To find the probability that the first tile selected is a consonant and the second tile is blank in the game of Scrabble, we follow these steps:
### Step 1: Determine the total number of tiles.
There are a total of 100 tiles.
### Step 2: Determine the number of consonant tiles.
From the problem, we know:
- Total number of consonant tiles = 56
### Step 3: Determine the number of blank tiles.
From the problem, we know:
- Total number of blank tiles = 2
### Step 4: Calculate the probability of selecting a consonant first.
The probability of selecting a consonant on the first draw is calculated as follows:
\[
P(\text{Consonant first}) = \frac{\text{Number of consonants}}{\text{Total number of tiles}} = \frac{56}{100} = 0.56
\]
### Step 5: Update the count of tiles after the first draw, considering that a consonant has been drawn.
After drawing one consonant tile, the number of available tiles is reduced from 100 to 99:
- Remaining consonants: 55 (since one has been drawn)
- Remaining blank tiles: 2 (not changed)
### Step 6: Calculate the probability of selecting a blank tile second.
The probability of selecting a blank tile after having drawn a consonant is:
\[
P(\text{Blank second | Consonant first}) = \frac{\text{Number of blank tiles}}{\text{Total remaining tiles}} = \frac{2}{99}
\]
### Step 7: Calculate the joint probability of both events happening.
The overall probability of both events (drawing a consonant first and then a blank second) is the product of the probabilities calculated in Steps 4 and 6:
\[
P(\text{Consonant first and Blank second}) = P(\text{Consonant first}) \times P(\text{Blank second | Consonant first})
\]
Substituting the values we have:
\[
P(\text{Consonant first and Blank second}) = \left(\frac{56}{100}\right) \times \left(\frac{2}{99}\right)
\]
Calculating this product gives:
\[
P(\text{Consonant first and Blank second}) = \frac{56 \times 2}{100 \times 99} = \frac{112}{9900}
\]
### Step 8: Simplify the fraction.
To simplify \(\frac{112}{9900}\), we can look for the greatest common divisor (GCD) of 112 and 9900.
Calculating or factoring gives:
- \(112 = 2^4 \times 7\)
- \(9900 = 2^2 \times 3^2 \times 5^2 \times 11\)
The GCD is \(4\) (as \(2^2\) is the common factor).
Now, simplifying:
\[
\frac{112 \div 4}{9900 \div 4} = \frac{28}{2475}
\]
### Step 9: Final answer or convert to a decimal or percentage.
To express \(\frac{28}{2475}\) as a decimal:
\[
\frac{28}{2475} \approx 0.0113
\]
To convert this to a percentage:
\[
0.0113 \times 100 \approx 1.13\%
\]
### Final Answers
- Exact Value: \(\frac{28}{2475}\)
- Decimal: Approximately 0.011 (rounded to 3 decimal places)
- Percentage: Approximately 1.1%
So, representing the final answer as a percentage, **the probability that the first tile selected is a consonant and the second tile is blank is approximately 1.1\%**.
### Step 1: Determine the total number of tiles.
There are a total of 100 tiles.
### Step 2: Determine the number of consonant tiles.
From the problem, we know:
- Total number of consonant tiles = 56
### Step 3: Determine the number of blank tiles.
From the problem, we know:
- Total number of blank tiles = 2
### Step 4: Calculate the probability of selecting a consonant first.
The probability of selecting a consonant on the first draw is calculated as follows:
\[
P(\text{Consonant first}) = \frac{\text{Number of consonants}}{\text{Total number of tiles}} = \frac{56}{100} = 0.56
\]
### Step 5: Update the count of tiles after the first draw, considering that a consonant has been drawn.
After drawing one consonant tile, the number of available tiles is reduced from 100 to 99:
- Remaining consonants: 55 (since one has been drawn)
- Remaining blank tiles: 2 (not changed)
### Step 6: Calculate the probability of selecting a blank tile second.
The probability of selecting a blank tile after having drawn a consonant is:
\[
P(\text{Blank second | Consonant first}) = \frac{\text{Number of blank tiles}}{\text{Total remaining tiles}} = \frac{2}{99}
\]
### Step 7: Calculate the joint probability of both events happening.
The overall probability of both events (drawing a consonant first and then a blank second) is the product of the probabilities calculated in Steps 4 and 6:
\[
P(\text{Consonant first and Blank second}) = P(\text{Consonant first}) \times P(\text{Blank second | Consonant first})
\]
Substituting the values we have:
\[
P(\text{Consonant first and Blank second}) = \left(\frac{56}{100}\right) \times \left(\frac{2}{99}\right)
\]
Calculating this product gives:
\[
P(\text{Consonant first and Blank second}) = \frac{56 \times 2}{100 \times 99} = \frac{112}{9900}
\]
### Step 8: Simplify the fraction.
To simplify \(\frac{112}{9900}\), we can look for the greatest common divisor (GCD) of 112 and 9900.
Calculating or factoring gives:
- \(112 = 2^4 \times 7\)
- \(9900 = 2^2 \times 3^2 \times 5^2 \times 11\)
The GCD is \(4\) (as \(2^2\) is the common factor).
Now, simplifying:
\[
\frac{112 \div 4}{9900 \div 4} = \frac{28}{2475}
\]
### Step 9: Final answer or convert to a decimal or percentage.
To express \(\frac{28}{2475}\) as a decimal:
\[
\frac{28}{2475} \approx 0.0113
\]
To convert this to a percentage:
\[
0.0113 \times 100 \approx 1.13\%
\]
### Final Answers
- Exact Value: \(\frac{28}{2475}\)
- Decimal: Approximately 0.011 (rounded to 3 decimal places)
- Percentage: Approximately 1.1%
So, representing the final answer as a percentage, **the probability that the first tile selected is a consonant and the second tile is blank is approximately 1.1\%**.