See a car

Problem:
If the probability of observing a car on a highway in 20 minutes time is 609/625 then what is the probability of observing a car in 5 minutes time on the same highway (considering all the factors involved to be uniform)?

Solution:
Highlight the part between the * symbols for the answer.
*Probability of seeing a car in 20 minutes = 609/625
=> Probability of not seeing a car in 20 minutes = 1 - 609/625 = 16/625
=> (Probability of not seeing a car in 5 minutes)^4 = 16/625
=> Probability of not seeing a car in 5 minutes = (16/625)^(1/4)
=> Probability of not seeing a car in 5 minutes = 2/5

Hence, the Probability of seeing a car in 5 minutes = 1 - 2/5 = 3/5*

Comments

  1. Unknown9:01 PM, July 23, 2020

    23/25

    ReplyDelete
  2. Unknown9:04 PM, July 23, 2020

    No, It will be 3/5.
    Idea:- Prob that no event in two half intervals = prob that there is no event in full interval.
    Therefore, 1-(1-p)^2=p' and 1-(1-p')^2=609/625. Where p is no event in the half intervals of first 10 minutes and p' is no event in the half interval of 20 minutes.
    Therefore, p'=21/25 and p=3/5.

    ReplyDelete
  3. Unknown9:05 PM, July 23, 2020

    It will be 3/5.
    Idea:- Prob that no event in two half intervals = prob that there is no event in full interval.
    Therefore, 1-(1-p)^2=p' and 1-(1-p')^2=609/625. Where p is no event in the half intervals of first 10 minutes and p' is no event in the half interval of 20 minutes.
    Therefore, p'=21/25 and p=3/5.

    ReplyDelete
  4. Pratik Poddar12:55 PM, September 08, 2026

    Correct - that recursion (halving the interval twice) is a nice way to do it. 1 - p' = 16/625 gives p' = 609/625... in your notation p' as "no event in 20 min halves": (16/625)^(1/2) = 4/25 for 10 minutes, then again (4/25)^(1/2) = 2/5 for 5 minutes, so the probability of seeing a car in 5 minutes is 1 - 2/5 = 3/5. Matches the posted solution. (replied using AI)

    ReplyDelete
  5. Pratik Poddar12:55 PM, September 08, 2026

    Agreed - 3/5 is right. (Your twin comment posted twice; see my reply on the other one for the cleaned-up algebra.) (replied using AI)

    ReplyDelete
  6. Pratik Poddar12:56 PM, September 08, 2026

    Not quite - the answer is 3/5. The slip is probably in splitting the interval: P(no car in 20 min) = 16/625, so P(no car in 5 min) = (16/625)^(1/4) = 2/5, hence P(see a car in 5 min) = 3/5. (You seem to have corrected yourself in the later comments - 3/5 it is.) (replied using AI)

    ReplyDelete

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