Construct the sample space for a probability experiment where a coin is flipped, and then a 10 -sided die is rolled.
\[
\{H 1, H 2, H 3, H 4, H 5, H 6, H 7, H 8, H 9, H 10, T 1, T 2, T 3, T 4, T 5, T 6, T 7, T 8, T 9, T 10\}
\]
This setup was used for a carnival game, and different amounts paid out for certain outcomes.
Find the probability of each outcome:
(a) There was a head on the coin and an odd number on the die.
(b) There was a head on the coin and a prime number on the die.
(c) There was a head on the coin and a number less than 9 on the die.
Write your answers in exact, simplified form.
Part 1 of 3
(a) The probability that there was a head on the coin and an odd number on the die is $\frac{1}{4}$.
Part: $1 / 3$
Part 2 of 3
(b) The probability that there was a head on the coin and a prime number on the die is
Skip Part Check
Answer
Final Answer: (a) The probability that there was a head on the coin and an odd number on the die is \(\boxed{\frac{1}{4}}\). (b) The probability that there was a head on the coin and a prime number on the die is \(\boxed{\frac{1}{5}}\). (c) The probability that there was a head on the coin and a number less than 9 on the die is \(\boxed{\frac{2}{5}}\).
Step 1 ：Construct the sample space for a probability experiment where a coin is flipped, and then a 10 -sided die is rolled. The sample space is \{H 1, H 2, H 3, H 4, H 5, H 6, H 7, H 8, H 9, H 10, T 1, T 2, T 3, T 4, T 5, T 6, T 7, T 8, T 9, T 10\}.
Step 2 ：The total number of outcomes is 20 (2 possibilities for the coin flip, heads or tails, and 10 possibilities for the die roll, 1 through 10).
Step 3 ：For part (a), the event is 'there was a head on the coin and an odd number on the die'. There are 5 ways this can occur (H1, H3, H5, H7, H9), so the probability is 5/20 = 1/4.
Step 4 ：For part (b), the event is 'there was a head on the coin and a prime number on the die'. The prime numbers less than 10 are 2, 3, 5, and 7. So there are 4 ways this can occur (H2, H3, H5, H7), so the probability is 4/20 = 1/5.
Step 5 ：For part (c), the event is 'there was a head on the coin and a number less than 9 on the die'. The numbers less than 9 are 1 through 8. So there are 8 ways this can occur (H1, H2, H3, H4, H5, H6, H7, H8), so the probability is 8/20 = 2/5.
Step 6 ：Final Answer: (a) The probability that there was a head on the coin and an odd number on the die is \(\boxed{\frac{1}{4}}\). (b) The probability that there was a head on the coin and a prime number on the die is \(\boxed{\frac{1}{5}}\). (c) The probability that there was a head on the coin and a number less than 9 on the die is \(\boxed{\frac{2}{5}}\).
