Question

you pick a card at random, put it back, and then pick another card at random.
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.

Explanation:

Step1: Determine the number of prime numbers

The numbers on the cards are \(3\), \(4\), \(5\). Prime numbers are numbers greater than \(1\) that have only two distinct positive divisors: \(1\) and itself. Among \(3\), \(4\), \(5\), the prime numbers are \(3\) and \(5\). So there are \(2\) prime numbers out of \(3\) cards.

Step2: Calculate the probability of picking a prime number in the first draw

The probability \(P_1\) of picking a prime number in the first draw is the number of prime - numbered cards divided by the total number of cards. Using the formula \(P=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of favorable outcomes and \(n(S)\) is the number of all possible outcomes. So \(P_1=\frac{2}{3}\).

Step3: Calculate the probability of picking a prime number in the second draw

Since the card is replaced, the total number of cards and the number of prime - numbered cards remain the same. So the probability \(P_2\) of picking a prime number in the second draw is also \(P_2 = \frac{2}{3}\).

Step4: Calculate the probability of both events (using the multiplication rule for independent events)

For two independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\). Here, \(P = P_1\times P_2\). Substitute \(P_1=\frac{2}{3}\) and \(P_2=\frac{2}{3}\) into the formula: \(P=\frac{2}{3}\times\frac{2}{3}=\frac{4}{9}\).

Step5: Convert the fraction to a percentage

To convert \(\frac{4}{9}\) to a percentage, use the formula \(Percentage=\frac{4}{9}\times100\%\). \(\frac{4}{9}\times100\%=\frac{400}{9}\%\approx44.4\%\)

Answer:

\(44.4\%\)