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A commuter passes through three traffic lights on her way to work. The chance that she will stop at all three lights is the same as the chance that none of the lights are red. The chance that two of the lights will be red is two times the chance that all the lights will be red. The chance that one light is red is $1/10$.

$X = \text{number of red lights for which the commuter stops on her way to work.} $

Question:

  1. What is the support of $X$?
  2. Give the PMF of $X$
  3. When the commuter drives to work, what is the probability she will not have to stop at any red lights?
  4. When the commuter drives to work, what is the probability that she has to stop for at least two red lights?

I found the the support of $X = \{0,1,2,3\}$

I am not sure how to find the PMF with the probabilities given.. any suggestions or hints?

2 Answers 2

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Your support is correct. To determine the pmf you need to use the information given, which is \begin{align*} P(X=0)&=P(X=3)\\ P(X=2)&=2P(X=3)\\ P(X=1) &= \frac{1}{10} \end{align*} We also use the fact that a pmf must sum to 1 over its support. \begin{align*} P(X=0)+P(X=1)+P(X=2)+P(X=3)&=1\\ \implies P(X=3)+P(X=1)+2P(X=3)+P(X=3)&=1\\ \implies 4P(X=3)+\frac{1}{10}&=1\\ \implies P(X=3)&=\frac{9}{40} \end{align*} Then $P(X=0)=\frac{9}{40}$ and $P(X=2)=\frac{18}{40}$ which gives you everything you need to state the pmf and answer your other questions.

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The chance that she will stop at all three lights is the same as the chance that none of the lights are red.

$\mathsf P(X=3)~=~\mathsf P(X=0)$

The chance that two of the lights will be red is two times the chance that all the lights will be red.

$\mathsf P(X=2) ~=~2~\mathsf P(X=3)$

The chance that one light is red is $1/10$

$\mathsf P(X=1)~=~\tfrac 1{10}$

So you have three simultaneous equations with four unknowns (one is trivial).   You need a fourth equation with those unknowns in order to obtain a unique solution.

$$\begin{align}p_0~&=~p_3 \\ p_2~&=~2p_3 \\ p_1 ~&=~ 1/10 \\ ?~&=~?\end{align}$$

Hint:

What do you know about total probability?