If 3 people are asked what day of the week they were born, find the probability that
i) 2 or 3 are the same.
ii) only 2 people are the same.
- is this 1/7 x 1/7 ??
If 4 people are asked what day of the week they were born, what is the probability that 2 or more people were born on the same day.
- How do I show this??
Using this idea, work out how many people need to be asked for their birthdate so that at least two people sharing their birthdate will have a chance of 1/2.
Thank you
i) It is the proportion below for 2 the same, plus 1/7*1/7*1/7 for 3 the same. (Either-or probabilities are determined by adding the separate probabilities.)
ii) It is 1/7*1/7*6/7, because the third person has a different day of the week.
iii) Work the same as i) above, for 2, 3 or 4 sharing the same day.
The only way I know to find out is to expand iii) above to larger numbers until it equals 1/2.
I hope this helps.
Sorry, I was distracted by your attempt to answer ii). Your concern is for matching the first throw, whatever it is.
i) It is the proportion below for 2 the same, plus 1/7*1/7 for 3 the same. (Either-or probabilities are determined by adding the separate probabilities.)
ii) The first throw can be any number, so your concern is with matching that number, so ii) = 1/7*6/7.
iii) Work the same as i) above, for 2, 3 or 4 sharing the same day.
The only way I know to find out is to expand iii) above to larger numbers until it equals 1/2.
To determine the probabilities, we can use the concept of combinations.
i) To find the probability that 2 or 3 people have the same day of the week birthday, we can calculate the total number of possible outcomes and the number of favorable outcomes.
Total number of outcomes:
Each person has 7 different days of the week to choose from when they were born. Therefore, for three people, the total number of outcomes is 7^3 = 343.
Number of favorable outcomes for two or three people having the same birthday:
There are 7 options for the shared birthday and for each of the remaining two people, they have 7 possible birthdays independently. So, the number of favorable outcomes is 7 * 7 * 7 = 343.
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 343 / 343 = 1
Therefore, the probability that 2 or 3 people share the same birthday is 1.
ii) To find the probability that exactly 2 people share the same birthday, we need to consider two scenarios: the first two people sharing the same birthday or the last two people sharing the same birthday.
Number of ways first two people share the same birthday:
There are 7 options for the shared birthday and for the remaining two people, they each have 6 possible birthdays (excluding the one already selected). So, the number of favorable outcomes is 7 * 6 * 6 = 252.
Number of ways last two people share the same birthday:
Same as above, there are 7 options for the shared birthday (excluding the one already selected). So, the number of favorable outcomes is 7 * 6 * 1 = 42.
Total number of outcomes:
Each person still has 7 different days of the week to choose from. Therefore, for three people, the total number of outcomes is 7^3 = 343.
Probability = Number of favorable outcomes / Total number of outcomes
Probability = (252 + 42) / 343 = 294 / 343 ≈ 0.857
Therefore, the probability that exactly 2 people share the same birthday is approximately 0.857.
Now, let's consider the case of 4 people.
To find the probability that 2 or more people have the same birthday, we can use the principle of complementary probabilities.
The probability that none of the 4 people share the same birthday can be calculated as follows:
First person has all 7 days to choose from.
Second person has 6 options (excluding the day chosen by the first person).
Third person has 5 options (excluding the days chosen by the first two people).
Fourth person has 4 options (excluding the days chosen by the first three people).
Therefore, the number of outcomes where each person has a different birthday is 7 * 6 * 5 * 4 = 840.
Total number of outcomes:
Each person has 7 different days of the week to choose from. Therefore, for four people, the total number of outcomes is 7^4 = 2401.
The probability of none of the 4 people sharing the same birthday is 840/2401 ≈ 0.35.
Hence, the probability that 2 or more people have the same birthday is 1 - 0.35 = 0.65 or approximately 65%.
Finally, let's determine the number of people needed for at least two people to have a 50% chance of sharing their birthday.
To calculate this, we can start with 1 person and increment the number of people until the probability reaches or exceeds 0.5.
Using the same logic as above, as the number of people increases, the probability that none of them share their birthday decreases. When we reach the point where the probability equals or exceeds 0.5, we know that at least two people must have the same birthday.
By trying different numbers of people, we find that when there are 23 people, the probability of at least two of them sharing a birthday is approximately 0.507. So, we need 23 people to have a chance of 1/2 (or 50%).
I hope this explanation helps!