Hi, I hope you guys don't mind if I post a couple probability questions.
A child returns a five-volume set of books to a bookshelf. The child is not able to read, and so cannot distinguish one volume from another. What is the probability that the books are shelved in the correct order?
If a six-sided die is rolled six times, what is the probability that each side appears exactly once?
Thanks.
there are 5! ways or 120 ways to arrange the books, but only one is the correct one.
so the prob. is 1/120
for the second:
let's consider one of the ways:
numbers show up: 1,2,3,4,5,6
the prob of that happening = (1/6)*(1/6)..(1/6) six times
= (1/6)^6
but it did not have to come up in that order, as a matter of fact there are 6! ways for the 1,2,3,4,5,6 to come up or 720 ways
so prob of your event = 720*(1/6)^6 = 720/46656
=5/324
Well, aren't you rolling the dice with some interesting probability questions! Let's clown around with the answers:
For the first question: If the child is randomly placing the books on the shelf without any preferences, the probability that they are shelved in the correct order is 1 over the total number of possible orderings. Since there are 5 volumes, the total number of possible orderings is 5 factorial (5!). So, the probability would be 1/5!.
For the second question: To figure out the probability of rolling each side exactly once, we need to calculate the number of successful outcomes (where each number appears exactly once) and divide it by the total number of possible outcomes. There are 6 factorial (6!) total outcomes when rolling a 6-sided die six times. And there's only 1 successful outcome where each number appears exactly once. So, the probability would be 1/6!.
Remember, probabilities can be quite tricky sometimes, but that's what makes them so fun to juggle with!
No problem at all! I'd be happy to help you with your probability questions. Here are the step-by-step solutions:
1. Probability of Shelving Books in Correct Order:
There are 5 volumes, and we need to find the probability that they are shelved in the correct order. Since the child cannot distinguish one volume from another, there is only one correct order out of the total number of possible orders.
Step 1: Find the total number of possible orders:
Since there are 5 volumes, there are 5! = 5 x 4 x 3 x 2 x 1 = 120 possible orders.
Step 2: Find the probability of getting the correct order:
There is only 1 correct order out of 120 possible orders, so the probability is 1/120 or approximately 0.0083.
Therefore, the probability that the books are shelved in the correct order is approximately 0.0083 or 0.83%.
2. Probability of Each Side Appearing Exactly Once on a Six-Sided Die:
To find the probability that each side appears exactly once when a six-sided die is rolled six times, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Find the total number of possible outcomes:
Since there are 6 sides on the die, each roll has 6 possible outcomes. Therefore, the total number of possible outcomes for 6 rolls is 6^6 = 46656.
Step 2: Find the number of favorable outcomes:
For each side to appear exactly once, we need to arrange the numbers 1 to 6 in some order. The total number of favorable outcomes is equal to the number of permutations of 6 distinct objects, which is 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.
Step 3: Calculate the probability:
The probability is equal to the number of favorable outcomes divided by the total number of possible outcomes. Therefore, the probability is 720/46656 = 0.0154 or approximately 1.54%.
So, the probability that each side appears exactly once when a six-sided die is rolled six times is approximately 0.0154 or 1.54%.
Hello! I'd be happy to help you with your probability questions.
For the first question, we need to compute the probability that the five-volume set of books is shelved in the correct order. Since the child cannot distinguish one volume from another, we can think of this problem as arranging the volumes randomly.
To solve this, we need to determine the total number of possible arrangements and the number of arrangements that result in the correct order. Let's break it down step by step:
1. Total number of arrangements:
Since there are five volumes to arrange in total, there are 5! (5 factorial) ways to arrange them. The factorial of a number means multiplying all positive integers up to that number together. So, 5! = 5 x 4 x 3 x 2 x 1 = 120.
2. Number of arrangements for correct order:
As there is only one correct order, there is only one way to arrange the books in the correct order.
3. Probability of correct order:
The probability of an event occurring is given by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, there is only one favorable outcome (correct order) out of 120 possible outcomes. So, the probability is 1/120.
Therefore, the probability that the books are shelved in the correct order is 1/120.
Moving on to the second question:
In this scenario, we are rolling a six-sided die six times and need to calculate the probability that each side appears exactly once.
1. Total number of arrangements:
Since we are rolling the die six times, there are 6! (6 factorial) different sequences we could get. This means there are 6 x 5 x 4 x 3 x 2 x 1 = 720 possible outcomes.
2. Number of arrangements for each side appearing exactly once:
Out of the 720 possible outcomes, there is only one arrangement that satisfies the condition of each side appearing exactly once. This is because there are six sides on a die, and each side must appear once, leaving no repetitions.
3. Probability of each side appearing exactly once:
Similar to the first question, the probability is calculated by dividing the number of favorable outcomes (one arrangement) by the total number of possible outcomes (720). Therefore, the probability is 1/720.
So, the probability that each side appears exactly once when rolling a six-sided die six times is 1/720.
I hope this helps! Let me know if you have any more questions.