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
what is the probability of picking a 2 and then picking a prime number?
write your answer as a decimal rounded to the nearest thousandth.

Explanation:

Step1: Calculate the probability of picking 2 first

There are 6 cards in total. The probability of picking card 2 is \(P(2)=\frac{1}{6}\)

Step2: Calculate the probability of picking a prime number second

After picking 2 (and not replacing it), there are 5 cards left. The prime - numbered cards among the remaining cards: 1 is not prime, 3 is prime, 4 is not prime, 5 is prime. So there are 2 prime - numbered cards left. The probability of picking a prime number given that 2 was picked first is \(P(\text{prime}|\text{2 picked})=\frac{2}{5}\)

Step3: Use the multiplication rule for dependent events

The multiplication rule for dependent events is \(P(A\cap B)=P(A)\times P(B|A)\). Here \(A\) is the event of picking 2 and \(B\) is the event of picking a prime number. So \(P = \frac{1}{6}\times\frac{2}{5}=\frac{2}{30}=\frac{1}{15}\approx0.067\)

Answer:

\(0.067\)