Question

Every day of spring the probability that a thunderstorm takes place in Toronto is 0,005. We assume that the events are independent and that the spring starts March 21st.

1) What date on average will produce the first spring storm in Toronto?

2) What is the probability that the first storm takes place April 25th?

3) What is the probability that the first storm takes place April 25th knowing that it is April 15th and there has not been a storm yet?

4) What is the probability that the third storm takes place May 1st?

5) What is the probability that the third storm takes place May 1st knowing that it is April 15th and a storm has just occurred the first time this Spring?

Answer 1

1) Let X be the random variable denote the number of days until first spring storm.

X ~ Geo(p), where, p = 0.05

So, E(X) = 1/p = 1/0.05

The average number of days to have the first thunderstorm is 20.

So, it will occur for the first time 20 days from March 21st, which is April 9th.

2) The probability that first storm takes place at 25th April = P(X = 36) = = 0.0083

3) The probability that first storm takes place on April 25th given that there is no storm till April 15th = P(X = 36 | X > 26)

= P(27 X 36) / P(X > 26) = 0.0879/0.2378 = 0.3696

4) The probability that 3rd storm takes place on May 1st = The probability there will be 2 storms between March 21st and April 30th = * 0.05 = 0.2773 * 0.05 = 0.0139