1. What is the probability that the first raindrop lands in the shaded region? H

Question

1. What is the probability that the first raindrop lands in the shaded region? Hint: View a raindrop as a single point in the triangle, What is the sample space? What is the event? How are events to be 2. What is the probability that both raindrops land in the shaded region? 3. What is the probability that no raindrops land in the shaded region? 4. What is the probability that at least one raindrop lands in the shaded region? 5. What is the probability that exactly one raindrop lands in the shaded region?

Explanation / Answer

1.

As the hint suggests, consider raindrop as a point in the triangle, now the sample space is the entire triangle, the event is the shaded region inside the traingle and we can measure the events by the areas of the entire triangle(sample space) and the shaded region(the event).

Now, clearly the dotted lines divide the triangle into 3 equal parts and out of these 3 parts 2 are shaded, Thus, Area of Shaded Region = (2/3)*(Area of triangle).

Now, Required probability = (Area of shaded region)/(Area of triangle) = 2/3.

2.

Now, we can assume the two raindrops to be independent. Thus:

Probability that both raindrops land in the shaded region = (Probability that first raindrop lands in shaded region)*(Probability that second raindrop lands in shaded region)

Now,

P(first raindrop lands in shaded region) = P(second raindrop lands in shaded region) = 2/3

Thus,

Probability that both raindrops land in the shaded region = 2/3*2/3 = 4/9.

3.

Now, we can assume the two raindrops to be independent. Thus:

Probability that no raindrops land in the shaded region = (Probability that first raindrop doesn't land in shaded region)*(Probability that second raindrop doesn't land in shaded region)

Now,

P(first raindrop doesn't land in shaded region) = 1 - P(first raindrop lands in shaded region) = 1 - 2/3 = 1/3

Similarly,

P(second raindrop doesn't land in shaded region) = 1/3

Thus,

Probability that no raindrops land in the shaded region = 1/3*1/3 = 1/9.

4.

P(atleast one raindrop lands in shaded region) = 1 - P(no raindrop lands in shaded region) = 1 - 1/9 = 8/9.

5.

P(exactly one raindrop lands in shaded region) = 1 - P(both raindrops land in shaded region) - P(no raindrops land in shaded region) = 1 - 4/9 - 1/9 = 4/9.

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