There are 8 streets to be named after 8 tree types. Ash, Birch, Cedar, Elm, Fir, Maple, Pine, and Willow. A city planner randomly selects the street names from the list of 8 tree types. Compute the probability of each of the following events. Event A: The first three streets are Willow, Maple, and Cedar, without regard to order. Event B: The first street is Pine, followed by Elm and then Ash.
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Given:
Total number of Streets = 8
Total Number of names=8
City planner randomly selects tree name.
To Find:
The probability of each of the following events.
Event A: The first three streets are Willow, Maple, and Cedar, without regard to order.
Event B: The first street is Pine, followed by Elm and then Ash.
Solution:
Since the names cant repeat,
- First street can be named in 8 ways
- Second in 7, third in 6 .. and last in one way.
- Total number of ways of naming = 8x7x6...x1 = 8!
Event A :The first three streets are Willow, Maple, and Cedar, without regard to order.
- Since order does not matter first 3 names can be choosen from the given 3 in 3x2x1 = 3! way.
- Remaining 5 can be arranged in 5! ways.
- Probablity of event A = 3! x 5!/ 8!
- P ( A ) = 6/6x7x8 = 1/56 .
Event B : The first street is Pine, followed by Elm and then Ash.
- Here the order is fixed.
- This fixed naming can be done only in 1 way.
- Remaining 5 can be arranged in 5! ways.
- Probability of event B = 5!/8!
- P ( B ) = 1/6x7x8 = 1/336
Logically , Event A will have more probability since it has more possible occurrences than Event B which is fixed.
The probability of event A = 1/56 and probability of event B = 1/336.
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