Axiomatic Approach to Probability

A sample space consists of $9$ elementary outcomes $e_1, e_2 ,\dots , e_9$ whose probabilities are

$P\left(e_1\right)=P\left(e_2\right)=.08, P\left(e_3\right)=P\left(e_4\right)=P\left(e_5\right)=.1$

$P\left(e_6\right)=P\left(e_7\right)=.2, P\left(e_8\right)=P\left(e_9\right)=.07$

Suppose $A=\left\{e_1, e_5, e_8\right\}, B=\left\{e_2, e_5, e_8, e_9\right\}$

Calculate $P\left(\overline{B}\right)$ from $P\left(B\right)$, also calculate $P\left(\overline{B}\right)$ directly from the elementary outcomes of $\overline{B}$

A sample space consists of $9$ elementary outcomes $e_1,e_2,\dots ,e_9$ whose probabilities are

$P\left(e_1\right)=P\left(e_2\right)=0.08,P\left(e_3\right)=P\left(e_4\right)=P\left(e_5\right)=0.1$

$P\left(e_6\right)=P\left(e_7\right)=0.2,P\left(e_8\right)=P\left(e_9\right)=0.07$

Suppose $A=\left\{e_1, e_5, e_8\right\},B=\left\{e_2, e_5, e_8, e_9\right\}$

List the composition of the event $A\cup B$, and calculate $P(A\cup B)$ by adding the probabilities of the elementary outcomes.

The accompanying Venn diagram shows three events, $A, B,$ and $C,$ and also the probabilities of the various intersections (for instance, $P(A\cap B)=0.07$). Determine $P(B\cap C)$

The accompanying Venn diagram shows three events, $A, B,$ and $C,$ and also the probabilities of the various intersections (for instance, $P(A\cap B)=.07$). Determine the probability of exactly one of the three occurs.

A sample space consists of $9$ elementary outcomes $e_1,e_2,\dots ,e_9$ whose probabilities are

$P\left(e_1\right)=P\left(e_2\right)=0.08,P\left(e_3\right)=P\left(e_4\right)=P\left(e_5\right)=0.1$

$P\left(e_6\right)=P\left(e_7\right)=0.2,P\left(e_8\right)=P\left(e_9\right)=0.07$

Suppose $A=\left\{e_1, e_5, e_8\right\},B=\left\{e_2, e_5, e_8, e_9\right\}$

Calculate $P\left(A\right),P\left(B\right)$ and $P(A\cap B)$.

The accompanying Venn diagram shows three events, $A, B,$ and $C,$ and also the probabilities of the various intersections (for instance, $P(A\cap B)=.07$). Determine $\mathrm{P}(\mathrm{A}\cup \mathrm{B})$

The accompanying Venn diagram shows three events, $A, B,$ and $C,$ and also the probabilities of the various intersections (for instance, $P(A\cap B)=.07$). Determine $P(B\cap \overline{C})$

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The accompanying Venn diagram shows three events, $A, B,$ and $C,$ and also the probabilities of the various intersections (for instance, $P(A\cap B)=.07$). Determine $P\left(A\right)$

Match the proposed probability under Column $C_1$ with the appropriate written description under column $C_2$

  $C_1$                                                                       $C_2$

Probability                                                    Written Description

$\left(a\right)$ $0.95$                                                 $\left(\mathrm{i}\right)$An incorrect assignment

$\left(b\right)$ $0.02$                                                 $\left(\mathrm{ii}\right)$ No chance of happening

$\left(c\right)$ $-0.3$                                                $\left(\mathrm{iii}\right)$ As much chance of happening as not.

$\left(d\right)$ $0.5$                                                    $\left(\mathrm{iv}\right)$ Very likely to happen

$\left(e\right)$ $0$                                                       $\left(\mathrm{v}\right)$ Very little chance of happening

One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently the sample space consists of four elementary outcomes $S=\{$ John promoted, Rita promoted, Aslam promoted, Gurpreet promoted$\}$. You are told that the chances of John's promotion is same as that of Gurpreet, Rita's chances of promotion are twice as likely as Johns. Aslam's chances are four times that of John.

Determine $P$(John promoted)
$P$(Rita promoted)
$P$(Aslam promoted)
$P$(Gurpreet promoted)

Four candidates $A, B, C, D$ have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as $B,$ and $B$ and $C$ are given about the same chance of being selected, while $C$ is twice as likely to be selected as $D,$ what are the probabilities that $A$ will not be selected? 

The probability of happening of an event A is $0.5$ and that of  $B$ is $0.3.$ If  $A$ and $B$ are mutually exclusive events, then the probability of neither $A$ or $B$ is

If $A$ and $B$ are two events associated with a random experiment such that $P\left(A\right)=0.3, P\left(B\right)=0.2$ and $P(A\cap B)=0.1$, then the value of $P(A\cap \overline{B})$ is _____.   [Write the answer in decimal form]

If $e_1, e_2, e_3, e_4$ are the four elementary outcomes in a sample space and $P\left(e_1\right)=.1, P\left(e_2\right)=.5, P\left(e_3\right)=.1$ then the probability of $e_4$ is _____.

The probability that the home team will win an upcoming football game is $0.77$, the probability that it will tie the game is $0.08$ and the probability that it will lose the game is _____.

Determine the probability $p$ for sum of $6$ appears in a single toss of a pair of fair dice.

Determine the probability $p$ for a king, $9$ of hearts, or $3$ of spades appears in drawing a single card from a well shuffled ordinary deck of $52$ cards.

Determine the probability $p$ for at least one head appears in two tosses of a fair coin.

Determine the probability $p$ for odd number appears in a single toss of a fair die.

A sample space consists of $9$ elementary outcomes $e_1,e_2,\dots ,e_9$ whose probabilities are

$P\left(e_1\right)=P\left(e_2\right)=0.08,P\left(e_3\right)=P\left(e_4\right)=P\left(e_5\right)=0.1$

$P\left(e_6\right)=P\left(e_7\right)=0.2,P\left(e_8\right)=P\left(e_9\right)=0.07$

Suppose $A=\left\{e_1, e_5, e_8\right\},B=\left\{e_2, e_5, e_8, e_9\right\}$

Using the addition law of probability, calculate $P(A\cup B)$.