Suppose that you toss a coin and roll a die. What is the probability of obtaining each of the following combinations? (Enter your answers as fractions.)
P(tails and 5) = 1
P(tails or 1) = 2
Suppose that you toss a coin and roll a die What is the probability of obtaining tails or a five? (Enter the probability as a fraction.)
Ah, probabilities and combinations, my two favorite subjects! Here we go:
P(tails and 5) = 1/12. Since we have a fair coin and a fair die, the probability of getting tails is 1/2 and the probability of rolling a 5 is 1/6. To find the probability of both events happening, we multiply the individual probabilities: (1/2) * (1/6) = 1/12. So, there's a 1/12 chance of landing on tails and rolling a 5.
P(tails or 1) = 2/3... is what some might say, but not me! Let's dive into it. The probability of getting tails is still 1/2, and the probability of rolling a 1 is 1/6. To find the probability of either event happening, we add the individual probabilities: (1/2) + (1/6) = 4/6 = 2/3. Voila! We have a 2/3 chance of landing on either tails or 1. That's a pretty decent chance, I'd say!
Remember, probabilities are all about figuring out the likelihood of events happening. And hey, even if it's not a sure thing, at least we can always rely on the fact that laughter is the best probability booster!
To calculate the probability of obtaining each of the given combinations, we need to determine the number of favorable outcomes and the total number of possible outcomes.
In this scenario, we toss a coin and roll a die, so let's consider the possible outcomes for each event:
For the coin toss, there are 2 possible outcomes: heads (H) or tails (T).
For the die roll, there are 6 possible outcomes: 1, 2, 3, 4, 5, or 6.
Now let's calculate the probabilities:
1. P(tails and 5):
To obtain tails (T) and 5, there is only one favorable outcome: Tails (T) and 5.
The total number of possible outcomes is the product of the number of outcomes for each event: 2 (coin toss) * 6 (die roll) = 12.
Therefore, the probability of obtaining tails and 5 is: P(Tails and 5) = 1/12.
2. P(tails or 1):
To obtain tails (T) or 1, we need to consider two scenarios: tails only and 1 only.
For tails only, there is one favorable outcome: Tails (T).
For 1 only, there is one favorable outcome: 1.
The total number of possible outcomes is still 12, as determined earlier.
To calculate the probability of tails or 1, we sum up the favorable outcomes for each scenario: 1 (tails only) + 1 (1 only) = 2.
Therefore, the probability of obtaining tails or 1 is: P(Tails or 1) = 2/12 = 1/6.
To find the probabilities of obtaining each of the given combinations, we need to first determine the total number of possible outcomes for the coin toss and the die roll.
1. P(tails and 5):
Since there is only one combination that satisfies both getting a tails on the coin toss and a 5 on the die roll, the probability is 1. Therefore, P(tails and 5) = 1.
2. P(tails or 1):
To calculate the probability of tails or 1, we need to determine the number of outcomes that satisfy either of the two conditions and divide it by the total number of possible outcomes.
For tails, there is only one possible outcome (tails), and for 1 on the die roll, there is only one possible outcome (rolling a 1). Since these two conditions are mutually exclusive (they cannot occur at the same time), we can add the number of outcomes that satisfy each condition.
Now, we need to calculate the total number of possible outcomes for the coin toss and the die roll.
For the coin toss, there are two possible outcomes (heads or tails).
For the die roll, there are six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
To find the total number of possible outcomes, we multiply the number of outcomes for each event:
Total outcomes = 2 (coin toss outcomes) * 6 (die roll outcomes) = 12.
Now, we can calculate the probability of tails or 1:
Number of outcomes that satisfy either condition = Number of tails outcomes + Number of 1 outcomes = 1 + 1 = 2.
P(tails or 1) = Number of outcomes that satisfy either condition / Total number of outcomes = 2 / 12 = 1/6.
Therefore, P(tails or 1) = 1/6.
P(tails and 5) = (1/2)*(1/6) = 1/12
P(tails OR 1 (but not both)) = (1/2)*(5/6)+(1/2)*(1/6) = 1/2