# Probability of Taxi Arrival in 10 Minutes After 1 Hour Wait

• mikemike123
In summary, the conversation is about finding the probability of a taxi arriving within the next 10 minutes at a busy intersection with an exponential distribution of 10 minutes between arrivals. The initial guess for the limits was (0<x<60) and (70<x<infinity), but it was incorrect. The correct approach is to find the conditional probability of Pr(60 < t < 70) given Pr(60 < t). The formula for this is needed to solve the problem.
mikemike123
Heres the question... The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.

The question I am stuck on is...

Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives within the next 10 minutes. (The first part of the problem was to find probability you wait longer then an hour which I figured the limits would be (60<x<infinity).

Well i know mew=beta=10 min=1/lambda=1/10

f(x)= lambda*e^-lambda which will ultimately give me 1/10e^-1/10xdx. I have my integral set up, the thing is I can't figure out my limits. My initial guess was to evaluate the integral from (0<x<60) and subtract (70<x<infinity), ultimately giving me the answer .9984 or 99.84%. I thought it was right but apparently wrong, can someone please help me set up the appropriate limits. Thanks in advance.

mikemike123 said:
Heres the question... The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.

The question I am stuck on is...

Suppose you have already been waiting for one hour for a taxi, what is the probability that one arrives within the next 10 minutes. (The first part of the problem was to find probability you wait longer then an hour which I figured the limits would be (60<x<infinity).

Well i know mew=beta=10 min=1/lambda=1/10

f(x)= lambda*e^-lambda which will ultimately give me 1/10e^-1/10xdx. I have my integral set up, the thing is I can't figure out my limits. My initial guess was to evaluate the integral from (0<x<60) and subtract (70<x<infinity), ultimately giving me the answer .9984 or 99.84%. I thought it was right but apparently wrong, can someone please help me set up the appropriate limits. Thanks in advance.

Not subtract. You have a conditional probability here.

Ok so if my given is the answer I got for my first part, .0025. How would I go on finding the Probability of P(AintersectB)?

mikemike123 said:
Ok so if my given is the answer I got for my first part, .0025. How would I go on finding the Probability of P(AintersectB)?

You need Pr(60 < t < 70) given Pr(60 < t). What's the formula for that?

## 1. What is the probability of a taxi arriving within 10 minutes after waiting for 1 hour?

The probability of a taxi arriving within 10 minutes after waiting for 1 hour depends on various factors such as the location, time of day, and demand for taxis. It is difficult to determine an exact probability as it can vary greatly.

## 2. How do you calculate the probability of a taxi arriving within 10 minutes after waiting for 1 hour?

The probability of a taxi arriving within 10 minutes after waiting for 1 hour can be calculated using the Poisson distribution formula. This formula takes into account the average number of taxis arriving per hour, which can be estimated based on historical data or current demand.

## 3. What are some factors that can affect the probability of a taxi arriving within 10 minutes after waiting for 1 hour?

The probability of a taxi arriving within 10 minutes after waiting for 1 hour can be affected by various factors, such as the location and time of day. High traffic areas or rush hour can decrease the probability, while less busy areas or off-peak times can increase it. Demand for taxis in the area can also impact the probability.

## 4. Is the probability of a taxi arriving within 10 minutes after waiting for 1 hour the same for all locations?

No, the probability of a taxi arriving within 10 minutes after waiting for 1 hour can vary depending on the location. Busy cities or areas with high demand for taxis may have a lower probability compared to suburban or rural areas with less demand.

## 5. Can the probability of a taxi arriving within 10 minutes after waiting for 1 hour be improved?

Yes, there are a few ways to potentially improve the probability of a taxi arriving within 10 minutes after waiting for 1 hour. One way is to use a taxi-hailing app that can connect you with available taxis in your area. Another way is to adjust your location or time of day to increase the chances of finding an available taxi. Additionally, pre-booking a taxi or using a ride-sharing service can also improve the probability of a timely arrival.

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