Question

When did you leave? The National Household Travel Survey gathers data on the time of day when people begin a trip in their car or other vehicle. Choose a trip at random and record the time at which the trip started. Here is an assignment of probabilities for the outcomes :

a. What probability should replace “?” in the table? Why?

b. Find the probability that the chosen trip did not begin between $9$A.M. and $12:59$P.M.

Short answer

Part(a) $0.218$ is the probability that should replace "?" in the table.

Part(b) $0.766$ is the probability that the chosen trip did not begin between $9$ A.M. and $12:59$ P.M.

Step-by-step solution

Part(a) Step 1 : Given information

We need to find the missing probability in given table.

Time of day	$10\mathrm{P}.\mathrm{M}-12.59\mathrm{A}.\mathrm{M}$	$1\mathrm{A}.\mathrm{M}-5.59\mathrm{A}.\mathrm{M}$	$6\mathrm{A}.\mathrm{M}-8.59\mathrm{A}.\mathrm{M}$	$9\mathrm{A}.\mathrm{M}-12.59\mathrm{P}.\mathrm{M}$
Probability	$0.040$	$0.033$	$0.144$	data-custom-editor="chemistry" $0.234$

Time of day	$1\mathrm{P}.\mathrm{M}-3.59\mathrm{P}.\mathrm{M}$	$4\mathrm{P}.\mathrm{M}-6.59\mathrm{P}.\mathrm{M}$	$7\mathrm{P}.\mathrm{M}-9.59\mathrm{P}.\mathrm{M}$
Probability	$0.208$	?	$0.123$

Part(a) Step 2 : Simplify

As we know, Sum of all probabilities should be $1$and probability should lie between $0$and $1$.

Now,

Sum of probabilities is $0.040+0.33+0.144+0.234+0.208+?+0.123=1$

Missing probability is $1-0.782=0.218$

Hence, $0.218$ should replace “?” in the table.

Part(b) Step 1 : Given information

We need to find probability that the chosen trip did not begin between $9$ A.M. and $12:59$ P.M.

Part(b) Step 2 : Simplify

Using complement rule,

$\mathrm{P}\left({\mathrm{A}}^{\mathrm{c}}\right)=\mathrm{P}(1-\mathrm{A})$

It is given that P( Between $9$A.M and $12:59$P.M) is $0.234$

Now,

P( Not Between $9$A.M and $12:59$ P.M) $=1-0.234=0.766$

Hence, $0.766$is the probability that the chosen trip did not begin between $9$A.M and $12:59$ P.M .