To calculate the probability of two independent events happening in sequence, you use the rule of multiplication for independent events. Here, the two events are picking a 2 (Event A) and then picking a 3 (Event B) from a set of cards.
Let's go through the procedure step-by-step:
1. Determine the probability of the first event (picking a 2). Assuming there is a standard deck of cards with digits from 0 to 9, there is one '2' card among 10 possible cards. Hence, the probability of picking a 2 on the first draw is 1 out of 10. Mathematically, this can be written as:
\[ P(A) = \frac{1}{10} \]
2. Determine the probability of the second event (picking a 3), given that the first card is put back (which means that the total number of cards remains the same). Just like with the 2 card, there is one '3' card among the 10 possible cards. Therefore, the probability of picking a 3 on the second draw is also 1 out of 10:
\[ P(B) = \frac{1}{10} \]
3. Since the first card is put back after the first draw, the draws are independent of each other; that is, the outcome of the first draw does not affect the outcome of the second draw.
4. To find the probability of both events A and B occurring in sequence, multiply the probabilities of each event happening independently.
\[ P(A \text{ and } B) = P(A) \times P(B) \]
\[ P(A \text{ and } B) = \frac{1}{10} \times \frac{1}{10} = \frac{1}{100} \]
Therefore, the probability of picking a 2 and then picking a 3 with replacement is \( \frac{1}{100} \), or 0.01 when expressed as a decimal.
