What is the probability of rolling a 5 on a 6-sided die?
I believe this is 1/6 because there is only one 5 on a 6-sided die.
I am stuck on this part of the question:
If you roll the die 6 times, how many 5’s do you expect to get? Would you definitely get this number of 5’s? Explain why or why not.
1/6 is correct
6 times, expect 1 five. But this is the average number. All you have to do is roll a die several times to see that it will not happen every time.
Q: What's the difference between theory and practice?
A: In theory, there is no difference, but in practice, there is!
Ah, rolling the dice, a classic game of probability! Now, let's tackle your question about rolling a 5 on a 6-sided die.
You are absolutely right that the probability of rolling a 5 on a single roll is 1/6. There's only one lucky number 5 out of the six possible outcomes.
Now, onto rolling the die six times. The expected number of 5's you would get can be calculated by multiplying the probability of rolling a 5 on one roll, which is 1/6, by the number of times you roll the die, which is 6. So, 1/6 multiplied by 6 gives us an expected value of 1.
However, hold onto your red rubber nose, because the key word here is "expected." This means that on average, over a large number of trials, you would expect about one 5 per six rolls. But it doesn't mean you'll always, without fail, get exactly one 5 every time you roll the die six times.
Probability is a quirky little mathematician, you see. There's always a chance you'll roll more or fewer 5's than expected. So, you might roll two 5's, or zero 5's, or even six 5's if luck is feeling particularly mischievous. The actual number of 5's you get can vary within the realm of possibilities, but on average, it will tend towards that expected value of one.
So, to recap, you can expect to get about one 5 if you roll the die six times, but clowns in probability land can be spontaneous, so don't be surprised if reality throws you a funny curveball now and then.
If you roll the 6-sided die 6 times, the number of 5's you expect to get can be calculated using the concept of expected value.
The probability of rolling a 5 on a single roll is indeed 1/6. To find the expected number of 5's, you can multiply the probability of getting a 5 on a single roll by the number of rolls.
Expected number of 5's = Probability of getting a 5 on a single roll * Number of rolls
Expected number of 5's = (1/6) * 6
Expected number of 5's = 1
Therefore, you would expect to get one 5 if you roll the 6-sided die six times.
However, it is important to note that the expected value is not a guarantee. It represents the average number of 5's you would expect to get in the long run. In any individual set of 6 rolls, you could get more or fewer 5's than the expected value. The actual number of 5's you get will vary due to the element of chance involved in rolling the die.
To determine the expected number of 5's when rolling a die 6 times, you need to multiply the probability of getting a 5 on a single roll (which is 1/6) by the number of rolls.
So, to find the expected number of 5's, you can calculate:
Expected number of 5's = (Probability of getting a 5 on each roll) × (Number of rolls)
In this case, the probability of getting a 5 on each roll is 1/6, and you are rolling the die 6 times. Therefore:
Expected number of 5's = (1/6) × 6 = 1
So, you would expect to get 1 5 when rolling the die 6 times.
However, it is important to note that this expected number is not a guarantee that you will actually get 1 5 in the 6 rolls. Probability is a measure of likelihood, and it represents the average outcome you would expect over a large number of trials.
In this specific case, while the expected number of 5's is 1, it is possible to get more or fewer 5's in any given set of 6 rolls. This is because each roll is an independent event, and the outcome of one roll does not affect the outcome of the next roll. So, even though the expected number is 1, there is a range of possible outcomes, including zero 5's, one 5, two 5's, and so on.
To be more certain of getting close to the expected number of 5's, you would need to increase the number of rolls and perform the experiment multiple times. The law of large numbers suggests that as the number of trials increases, the observed results should approach the expected probability.