Question

Our class has 50 students. Assuming that no students are born on leap days (February29), what is the probability that no two students share the same birthday? What is the probability that at least one of the students has the same birthday as another student in the class? Please provide your answers in the form of a fraction.

Answer 1

Answer:  a) $\dfrac{799}{31250}$  b) $\dfrac{30451}{31250}$

Step-by-step explanation:

Since we have given that

Number of days in a year = 365

Number of students = 50

Probability that no two students share the same birthday. It means each student has different birthday.  

Probability(no two students share the same birthday)$=\dfrac{365-1}{365}\times \dfrac{365-2}{365}........\times \dfrac{365-50}{365}$

Probability(no two students share the same birthday)$=\dfrac{^{365}P_{50}}{365^{50}}$

Probability(no two students share the same birthday)$=\dfrac{799}{31250}\approx 0.025568$

Probability that at least one of the students has the same birthday as another student in the class is given by

$1-P(\text{no student share the same birthday})\\\\=1-\dfrac{799}{31250}\\\\=\dfrac{30451}{31250}=0.974432$