A math teacher observes that on the average 3 students came to their 8-9AM class late. The probability 5 students will be late in their 8-9AM class is ???
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To find the probability that exactly 5 students will be late in the 8-9AM class, we need to use the concept of a Poisson distribution. In this case, the average number of students coming late to the class is given as 3.
The Poisson distribution formula for the probability of a certain number of events occurring in a fixed interval of time when the average rate of occurrence is known is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
P(x; λ) is the probability of x events occurring,
e is the mathematical constant (approximately equal to 2.71828),
λ is the average rate of occurrence (in this case, 3),
x! represents the factorial of x.
Using this formula, we can calculate the probability of exactly 5 students being late in the 8-9AM class. Plugging in the values, we have:
P(5; 3) = (e^(-3) * 3^5) / 5!
Now, let's calculate this probability step by step:
First, calculate e^(-3):
e^(-3) = 0.04978706837 (rounded to 11 decimal places)
Next, calculate 3^5:
3^5 = 243
Now, calculate 5! (5 factorial):
5! = 5 * 4 * 3 * 2 * 1 = 120
Finally, plug in the calculated values:
P(5; 3) = (0.04978706837 * 243) / 120
Calculating this expression, we get:
P(5; 3) ≈ 0.10082
Therefore, the probability that exactly 5 students will be late in the 8-9AM class is approximately 0.10082 or 10.082%.