To find the probability that a die shows a number less than 5 and a coin shows a head, we need to consider the probabilities of each event separately and then combine them.

First, let's consider the die. A standard die has 6 faces numbered from 1 to 6. The event that the die shows a number less than 5 includes the numbers 1, 2, 3, and 4. So, there are 4 favorable outcomes for the die:

$1, 2, 3, 4$

The probability $P(\text{die shows less than 5})$ is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:

$P(\text{die shows less than 5}) = \frac{4}{6} = \frac{2}{3}$

Next, let's consider the coin. A standard coin has two faces: heads and tails. The probability $P(\text{coin shows head})$ of getting a head is:

$P(\text{coin shows head}) = \frac{1}{2}$

The events of rolling the die and tossing the coin are independent, so we can find the combined probability by multiplying the probabilities of each individual event:

$P(\text{die < 5 and coin = head}) = P(\text{die shows less than 5}) \times P(\text{coin shows head}) = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} = \frac{1}{3}$

Therefore, the probability that the die shows a number less than 5 and the coin shows a head is $\frac{1}{3}$.

Would you like further details or have any questions about this solution?

Here are 8 relative questions that expand on this topic:

1. What is the probability of rolling a number greater than 4 on a die and getting a tail on the coin?
2. If two coins are tossed, what is the probability of getting at least one head?
3. What is the probability of rolling an even number on the die and getting a head on the coin?
4. How does the probability change if the die is biased such that the number 6 is twice as likely as the other numbers?
5. What is the probability of rolling a 5 or 6 on the die and getting a head on the coin?
6. If two dice are rolled and one coin is tossed, what is the probability of getting a sum of 7 on the dice and a head on the coin?
7. What is the probability of rolling a number less than 3 on the die and getting a tail on the coin?
8. How would the probability change if the coin is biased and shows heads 60% of the time?

Tip: When dealing with multiple events, consider whether they are independent or dependent to determine the appropriate method for finding the combined probability.