Notes

To find the probability of rolling an odd number on a dice and getting tails on a coin toss (P(odd number, tails)), you need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down:

1. **Odd numbers on a dice:**
- The odd numbers on a standard six-sided dice are 1, 3, and 5.

2. **Coin toss:**
- There are two possible outcomes for a coin toss: heads or tails.

Now, let's find the favorable outcomes:

- If you roll a 1, 3, or 5 on the dice and get tails on the coin toss.

These are the three favorable outcomes.

Next, let's find the total number of possible outcomes:

- There are 3 odd numbers on a six-sided dice (1, 3, 5).
- There are 2 outcomes for the coin toss (heads or tails).

So, the total number of possible outcomes is 3 (odd numbers) * 2 (coin toss outcomes) = 6.

Now, the probability (P) is given by the formula:

[ P(text{odd number, tails}) = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} ]

[ P(text{odd number, tails}) = frac{3}{6} ]

Simplify the fraction:

[ P(text{odd number, tails}) = frac{1}{2} ]

So, the probability of rolling an odd number on a dice and getting tails on a coin toss is ( frac{1}{2} ).

To find the probability of rolling a number less than 5 on a dice and getting heads on a coin toss (P(<5, heads)), you need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down:

1. **Numbers less than 5 on a dice:**
- The numbers less than 5 on a standard six-sided dice are 1, 2, 3, and 4.

2. **Coin toss:**
- There are two possible outcomes for a coin toss: heads or tails.

Now, let's find the favorable outcomes:

- If you roll a 1, 2, 3, or 4 on the dice and get heads on the coin toss.

These are the four favorable outcomes.

Next, let's find the total number of possible outcomes:

- There are 4 numbers less than 5 on a six-sided dice (1, 2, 3, 4).
- There are 2 outcomes for the coin toss (heads or tails).

So, the total number of possible outcomes is 4 (numbers less than 5) * 2 (coin toss outcomes) = 8.

Now, the probability (P) is given by the formula:

[ P(<5, heads) = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} ]

[ P(<5, heads) = frac{4}{8} ]

Simplify the fraction:

[ P(<5, heads) = frac{1}{2} ]

So, the probability of rolling a number less than 5 on a dice and getting heads on a coin toss is ( frac{1}{2} ).

To find the probability of rolling an even number on a dice and getting heads on a coin toss (P(even number, heads)), you need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down:

1. **Even numbers on a dice:**
- The even numbers on a standard six-sided dice are 2, 4, and 6.

2. **Coin toss:**
- There are two possible outcomes for a coin toss: heads or tails.

Now, let's find the favorable outcomes:

- If you roll a 2, 4, or 6 on the dice and get heads on the coin toss.

These are the three favorable outcomes.

Next, let's find the total number of possible outcomes:

- There are 3 even numbers on a six-sided dice (2, 4, 6).
- There are 2 outcomes for the coin toss (heads or tails).

So, the total number of possible outcomes is 3 (even numbers) * 2 (coin toss outcomes) = 6.

Now, the probability (P) is given by the formula:

[ P(text{even number, heads}) = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} ]

[ P(text{even number, heads}) = frac{3}{6} ]

Simplify the fraction:

[ P(text{even number, heads}) = frac{1}{2} ]

So, the probability of rolling an even number on a dice and getting heads on a coin toss is ( frac{1}{2} ).

To find the probability of rolling a number greater than 2 on a dice and getting tails on a coin toss (P(>2, tails)), you need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down:

1. **Numbers greater than 2 on a dice:**
- The numbers greater than 2 on a standard six-sided dice are 3, 4, 5, and 6.

2. **Coin toss:**
- There are two possible outcomes for a coin toss: heads or tails.

Now, let's find the favorable outcomes:

- If you roll a 3, 4, 5, or 6 on the dice and get tails on the coin toss.

These are the four favorable outcomes.

Next, let's find the total number of possible outcomes:

- There are 4 numbers greater than 2 on a six-sided dice (3, 4, 5, 6).
- There are 2 outcomes for the coin toss (heads or tails).

So, the total number of possible outcomes is 4 (numbers greater than 2) * 2 (coin toss outcomes) = 8.

Now, the probability (P) is given by the formula:

[ P(>2, tails) = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} ]

[ P(>2, tails) = frac{4}{8} ]

Simplify the fraction:

[ P(>2, tails) = frac{1}{2} ]

So, the probability of rolling a number greater than 2 on a dice and getting tails on a coin toss is ( frac{1}{2} ).

To find the probability of rolling a prime number on a dice and getting tails on a coin toss, you first need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's break it down:

1. **Prime numbers on a dice:**
- The prime numbers on a standard six-sided dice are 2, 3, and 5.

2. **Coin toss:**
- There are two possible outcomes for a coin toss: heads or tails.

Now, let's find the favorable outcomes:

- If you roll a 2 on the dice and get tails on the coin toss.
- If you roll a 3 on the dice and get tails on the coin toss.
- If you roll a 5 on the dice and get tails on the coin toss.

These are the three favorable outcomes.

Next, let's find the total number of possible outcomes:

- There are 3 prime numbers on a six-sided dice (2, 3, 5).
- There are 2 outcomes for the coin toss (heads or tails).

So, the total number of possible outcomes is 3 (prime numbers) * 2 (coin toss outcomes) = 6.

Now, the probability (P) is given by the formula:

[ P(text{prime number, tails}) = frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}} ]

[ P(text{prime number, tails}) = frac{3}{6} ]

Simplify the fraction:

[ P(text{prime number, tails}) = frac{1}{2} ]

So, the probability of rolling a prime number on a dice and getting tails on a coin toss is ( frac{1}{2} ).

When a dice is rolled and a coin is tossed, there are a few possible outcomes, depending on the number of sides on the dice and the type of coin. Let's consider a standard six-sided dice and a fair coin with two sides (heads and tails).

For the dice roll:
- The dice has six faces numbered 1 through 6.
- The outcome of the dice roll could be any one of these numbers: 1, 2, 3, 4, 5, or 6.

For the coin toss:
- The coin has two sides: heads and tails.
- The outcome of the coin toss could be either heads or tails.

Combining the outcomes of the dice roll and the coin toss, you have a set of possible pairs. For example:
1. Dice roll: 1, Coin toss: Heads
2. Dice roll: 1, Coin toss: Tails
3. Dice roll: 2, Coin toss: Heads
4. Dice roll: 2, Coin toss: Tails
5. Dice roll: 3, Coin toss: Heads
6. Dice roll: 3, Coin toss: Tails
7. Dice roll: 4, Coin toss: Heads
8. Dice roll: 4, Coin toss: Tails
9. Dice roll: 5, Coin toss: Heads
10. Dice roll: 5, Coin toss: Tails
11. Dice roll: 6, Coin toss: Heads
12. Dice roll: 6, Coin toss: Tails

There are a total of 12 possible outcomes when rolling the dice and tossing the coin. Each outcome has an equal probability of 1/12 if both the dice and the coin are fair.