Event probability in Laplace experiments

The calculation of event probabilities in Laplace experiments is quite simple. Assume that a Laplace experiment has $m$ outcomes, and event $E$ contains $k$ of those outcomes, say $E=\{o_1,o_2, ..., o_k\}$ .

As each outcome has the probability $p=1/m$ to occur, and the probability of an event is the sum of its outcome probabilities, we get

\begin{array}{lll} p(E)&=& p(o_1)+...+p(o_k)\\[0.2em] &=& \frac{1}{m}+...+\frac{1}{m}\\[0.2em] &=& \frac{k}{m} \end{array}

Thus, we have the following theorem:

Theorem 1

Consider a Laplace experiment with $m$ possible outcomes and an event $E$ which contains $k$ of those outcomes. Then

p(E)=\frac{k}{m}

Using set notation, we have $\vert S\vert =m$ , where $S$ is the sample space, and $\vert E\vert =k$ , so we can rewrite the formula above as

p(E)=\frac{\vert E\vert}{\vert S\vert}

And if we think of the outcomes in $E$ as "desired", we can write

p(E)=\frac{\text{num. of desired outcomes}}{\text{num. of possible outcomes}}

Apply the formula to the following problems. Important: always consider whether we really have a Laplace experiment before us. Otherwise the above formula does not apply.

Exercise 1

Q1

A fair die is rolled once. Determine the probability that the resulting number is even.

Q2

A fair coin is tossed twice. Determine the probability for at least one head.

Q3

A fair die is rolled twice. Determine the probability that the sum of the two observed numbers is between $4$ and $7$ (including $4$ and $7$ ).

Q4

In a group of $82$ people, $10$ are tall. A person is selected at random, where each person can be chosen with the same probability. Determine the probability that you select a tall person.

Q5

In a class of $25$ students, everyone plays chess, tennis, or nothing. It is found that $6$ of the students play both tennis and chess, $10$ play tennis only, and $3$ play nothing. A student is selected at random from the group, where each student can be chosen with the same probability. Find the probability that the student

plays both tennis and chess.
plays chess only.
does not play chess.

Q6

In a group of people, $72\%$ have green eyes. A person is selected at random, where each person can be chosen with the same probability. What is the probability that this person has green eyes?

Q7

A town has $3$ newspapers. $20\%$ of the population read newspaper A (and perhaps other newspapers), $16\%$ read newspaper B (and perhaps other newspapers), and $14\%$ read newspaper C (and perhaps other newspapers). $8\%$ read newspapers A and B (and perhaps another one), $5\%$ read newspapers A and C (and perhaps another one), and $4\%$ read newspapers B and C (and perhaps another one). $4\%$ read all three newspapers, the rest does not read at all.

A person is selected at random from the town, where each person can be chosen with the same probability.

What is the probability that the person reads exactly one newspaper?
What is the probability that the person does not read any newspaper?

Solution

A1

Laplace experiment with $S=\{1,2,3,4,5,6\}$ and $E=\{2,4,6\}$ . Thus $p(E)=\frac{|E|}{|S|}=\frac{3}{6}=\underline{0.5}$ .

A2

Laplace experiment with $S=\{HH, HT, TH, TT\}$ and $E=\{HH,HT,TH\}$ . Thus $p(E)=\frac{|E|}{|S|}=\frac{3}{4}=\underline{0.75}$

A3

Laplace experiment with $\vert S\vert =36$ and $E=18$ (see figure below). Thus, $p(E)=\frac{18}{36}=\underline{\frac{1}{2}}$

A4

It is a Laplace experiment, as each person has the same probability to be selected. The possible outcomes are the people, so $S$ ="the people"= $\{P_1,...,P_{82}\}$ , $\vert S\vert=82$ . Assume that the tall ones are persons $P_1,..., P_{10}$ , so $E=\{P_1,...,P_{10}\}$ , and thus $\vert E\vert = 10$ . It follows $p(E)=\frac{\vert E\vert}{\vert S\vert}=\underline{\frac{10}{82}}$

A5

$S$ ="the students"= $\{s_1,...,s_{25}\}$ , $\vert S\vert=25$ . Each outcome (a student) has the same probability to occur $\rightarrow$ Laplace experiment. Event $T$ ="students playing tennis" (e.g. $T=\{s_1,s_4, ...\}$ ), event $C$ ="students playing chess" (e.g. $C=\{s_1,s_2, ...\}$ ). See figure below.

$p=\frac{\vert T\cap C\vert}{\vert S\vert }=\underline{\frac{6}{25}}$
$p=\underline{\frac{6}{25}}$
$p=\underline{\frac{13}{25}}$

A6

Laplace experiment, and $E$ ="green eyes". Assume, for example, that there are $100$ people in the group, so $72$ will have green eyes, so $\vert E\vert =72$ , and $p(E)=\frac{\vert E\vert }{\vert S\vert} = \frac{72}{100}=0.72$ . You can assume any other number of people in the group, and you will still get $p(E)=\underline{0.72}$ .

A7

Draw a Venn-diagram, and indicate the probabilities, starting with the $4\%$ (see figure below).

$p(\text{"1 newspaper"})=11\%+8\%+9\%=\underline{28\%}$
$p(\text{"0 newspaper"})=100\%-11\%-8\%-9\%-4\%-1\%-4\%=\underline{63\%}$