Question

Assume there are 365 days in a year.

1. What is the probability that ten students in a class have different birthdays?
2. What is the probability that at least ten students in a class share their birthday?

Answer 1

1# 30 to 60%

2# And 1 to 5%

Answer 2

Answer with Explanation:

1.

Number of days in a year= 365

we have to find the probability that ten students in a class have different birthdays.

Assuming that there are 30 days in month,and there are 12 months.

Number of ways of choosing 10 different days from 30 days is an arrangement of 10 things selected from 30  things when order of the arrangement is important and remaining 5 days can be arranged in such a way that there are 5 box and 10 different balls have to be placed

Total Possible outcome $=_{10}^{365}\textrm{C}$

Total favorable Outcome $=_{10}^{30}\textrm{P}*12+5^{10}$

Required Probability

  $=\frac{\text{Total favorable Outcome}}{\text{Total Possible Outcome}}\\\\=\frac{_{10}^{30}\textrm{P}*12+5^{10}}{_{10}^{365}\textrm{C}}$

$=\frac{1.09027350430*10^{14}+5^{10}}{1.02129640405931191878*10^{19}}$

=0.000010675(approx)

2.  At least ten students in a class share their birthday on the same day.

So, from 365 days we have to chose a single day which can be chosen in 365  ways.

And, total possible outcome is also 365.

Required Probability  

 $=\frac{\text{Total favorable Outcome}}{\text{Total Possible Outcome}}\\\\=\frac{_{1}^{365}\textrm{C}}{_{1}^{365}\textrm{C}}$

= 1 Way