A coin is tossed and then a die is rolled. Find the probability of getting a 5 on the die given that the coin landed tails up.
Well, let's see. If the coin landed tails up, it means it's not a head... so it's a tail! And with a die, there are 6 possible outcomes, right? So the probability of getting a 5 on the die, given that the coin landed tails up, is... drumroll, please... 1 out of 6! Because tails up or not, it's still a fair die and every outcome has an equal chance. Good luck rolling that 5, my friend!
To find the probability of getting a 5 on the die given that the coin landed tails up, we need to consider the sample space and the favorable outcomes.
Step 1: Determine the sample space:
The sample space is the set of all possible outcomes of tossing a coin and rolling a die. For a coin toss, there are 2 possible outcomes (heads or tails), and for rolling a die, there are 6 possible outcomes (numbers 1 to 6). So, the sample space is 2 (coin) × 6 (die) = 12.
Step 2: Determine the favorable outcomes:
We want to find the probability of getting a 5 on the die, given that the coin landed tails up. The coin landing tails up restricts the sample space to only the outcomes where the coin landed tails up. There are 2 possible outcomes for the coin (heads or tails), but we are only interested in the case where the coin landed tails up. So, the favorable outcomes in this case are the ones where the die shows a 5 when the coin has landed tails up. There is only 1 favorable outcome, which is the die showing a 5.
Step 3: Calculate the probability:
The probability is defined as the number of favorable outcomes divided by the number of possible outcomes.
Number of favorable outcomes = 1
Number of possible outcomes given that the coin landed tails up = 6 (number of possible outcomes on the die)
Therefore, the probability of getting a 5 on the die given that the coin landed tails up is 1/6.
To find the probability of getting a 5 on the die given that the coin landed tails up, we need to consider the sample space and the possible outcomes.
Step 1: Determine the sample space.
When a coin is tossed, it can either land heads or tails, so the sample space is {H, T}.
When a die is rolled, it can land any number from 1 to 6, so the sample space is {1, 2, 3, 4, 5, 6}.
Step 2: Determine the favorable outcomes.
We are given that the coin landed tails up, so we only need to consider outcomes where the coin lands tails up. The favorable outcomes are {T1, T2, T3, T4, T5, T6}.
Step 3: Calculate the probability.
The probability of an event occurring is defined as the number of favorable outcomes divided by the total number of possible outcomes.
In this case, the number of favorable outcomes is 6 (T1, T2, T3, T4, T5, T6), and the total number of possible outcomes is 2 (H, T) for the coin multiplied by 6 (1, 2, 3, 4, 5, 6) for the die, which gives us 12.
Therefore, the probability of getting a 5 on the die given that the coin landed tails up is 6/12, which simplifies to 1/2 or 0.5.
So the probability is 0.5, which means there is a 50% chance of rolling a 5 on the die when the coin lands tails up.
This is an example of conditional probability
P(A│B) read: the probabiltiy of B, given A
= P(A and B)/P(B)
P(A and B) = (1/6)(1/2) = 1/12
P(B) = 1/2
so P(A│B) = (1/12)/(1/2) = 1/6