Question

You pick a card at random. Without putting the first card back, you pick a second card at random. 1 2 3 4 5 6 7 8 What is the probability of picking a prime number and then picking a prime number? Write your answer as a percentage rounded to the nearest tenth.

Answer 1

To determine the probability of picking two prime numbers consecutively from the numbers 1 to 8, we first identify the prime numbers in this range.

The prime numbers between 1 and 8 are: 2, 3, 5, and 7.

So, the prime numbers are \( \{2, 3, 5, 7\} \). There are a total of 4 prime numbers.

Now, let's calculate the total number of cards, which is 8 (the numbers 1 through 8).

**Step 1: Calculate the probability of picking a prime number on the first draw.**
The probability of drawing a prime number first is:

\[
P(\text{First prime}) = \frac{\text{Number of prime numbers}}{\text{Total numbers}} = \frac{4}{8} = \frac{1}{2}
\]

**Step 2: Calculate the probability of picking another prime number on the second draw.**
If we have picked a prime number first, there are now 3 prime numbers left (since we don't replace the first card) and a total of 7 cards remaining.

The probability of drawing another prime number after the first is:

\[
P(\text{Second prime | First prime}) = \frac{\text{Remaining prime numbers}}{\text{Remaining total numbers}} = \frac{3}{7}
\]

**Step 3: Calculate the combined probability of both events.**
To find the overall probability of both events happening (picking a prime number first and then a prime number second), we multiply the probabilities of each step:

\[
P(\text{First prime and Second prime}) = P(\text{First prime}) \times P(\text{Second prime | First prime}) = \frac{4}{8} \times \frac{3}{7} = \frac{1}{2} \times \frac{3}{7} = \frac{3}{14}
\]

**Step 4: Convert the probability to percentage.**
To convert \(\frac{3}{14}\) to a percentage, we can perform the division:

\[
\frac{3}{14} \approx 0.2142857
\]

Now, to express it as a percentage:

\[
0.2142857 \times 100 \approx 21.42857
\]

Rounding this to the nearest tenth gives:

\[
21.4\%
\]

Thus, the probability of picking a prime number and then picking another prime number, rounded to the nearest tenth, is:

\[
\boxed{21.4\%}
\]