We are given the set of cards: $3, 4, 5, 6, 7, 8$, and asked to find the probability of picking an odd number first, followed by an even number.

Step 1: Identify the odd and even numbers

• Odd numbers: $3, 5, 7$ (3 odd numbers)
• Even numbers: $4, 6, 8$ (3 even numbers)

Step 2: Probability of picking an odd number first

There are 6 cards in total, and 3 of them are odd. So, the probability of picking an odd number is:

$P(\text{odd first}) = \frac{3}{6} = \frac{1}{2}$

Step 3: Probability of picking an even number second

After picking the first card (which was odd), there are now 5 cards remaining. Out of these, 3 are even. So, the probability of picking an even number second is:

$P(\text{even second}) = \frac{3}{5}$

Step 4: Multiply the probabilities

Since these are independent events (the outcome of one event does not affect the other), the total probability is the product of the two probabilities:

$P(\text{odd first and even second}) = P(\text{odd first}) \times P(\text{even second}) = \frac{1}{2} \times \frac{3}{5} = \frac{3}{10}$

Step 5: Convert to a percentage

To express the probability as a percentage, multiply by 100:

$P(\text{odd first and even second}) = \frac{3}{10} \times 100 = 30\%$

Thus, the probability of picking an odd number first and an even number second is 30%.

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Would you like more details or have any questions? Here are some related questions:

1. What is the probability of picking two even numbers in a row?
2. If you pick two cards at random with replacement, what is the probability of picking an odd number followed by an even number?
3. How would the probability change if there were different numbers of odd and even cards?
4. What is the probability of picking two odd numbers in a row from this set?
5. How do you calculate the probability of compound events in general?

Tip: When working with probabilities of compound events, always ensure that you adjust the denominator for events where the sample space changes (like when you don't replace the first card).