The sample space for the experiment in question is shown in the table below. $\mathrm{S}=\left{\begin{array}{c}(3,1),(3,2),(3,3),(3,4),(3,5),(3,6), \ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) \ 1...
Given that the two numbers appearing on throwing two dice are different. Find the probability of the event ‘the sum of numbers on the dice is 4 ‘.
Solution: Let $\mathrm{E}$ be the event that 'the sum of numbers on the dice is $4^{\prime}$ and $\mathrm{F}$ be the event that 'the two numbers appearing on throwing the two dice are different. So...
An instructor has a question bank consisting of 300 easy True / False questions, 200 difficult True / False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple-choice question?
Solution: There are two sorts of questions in this section: True/False and Multiple Choice Questions (T/F or MCQ), and each of these types is further subdivided into Easy and Difficult categories,...
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?
Solution Assume $B$ denotes boy and $G$ denote girl. Then, the sample space of the given experiment is $S={G G, G B, B G, B B}$ Let $\mathrm{E}$ be the event that 'both are girls'. $\Rightarrow...
A fair die is rolled. Consider events
and
. Find (i)
and
(ii)
and
(iii)
and ![Rendered by QuickLaTeX.com P((E \cap F) \mid G)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-77baf03a0ec38335ef5d997d412cf8f0_l3.png)
Solution: The sample space for the given event is $S={1,2,3,4,5,6}$ Sample space of events,$E={1,3,5}, F={2,3}$ and $G={2,3,4,5} \ldots \ldots \ldots$ (i)The events probability of events are:...
A black and a red dice are rolled.(a) Find the conditional probability of obtaining a sum greater than 9 , given that the black die resulted in a
(b) Find the conditional probability of obtaining the sum 8 , given that the red die resulted in a number less than ![Rendered by QuickLaTeX.com 4 .](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-a0b9d4c10d0939a21cebbdfe1d0cbac7_l3.png)
Solution:Assuming: B denote black coloured die and R denote red coloured die. Therefore, the sample space for the specified experiment will be as follows:...
Mother, father and son line up at random for a family picture E: son on one end, F: father in middle
Solution: We assume that $\mathrm{M}$ denote mother, $\mathrm{F}$ denote father and $\mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= { MFS, SFM, FSM, MSF, SMF,...
Mother, father and son line up at random for a family picture E: son on one end, F: father in middle
Solution: We assume that $mathrm{M}$ denote mother, $mathrm{F}$ denote father and $mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= $\{$ MFS, SFM, FSM, MSF, SMF,...
Mother, father and son line up at random for a family picture E: son on one end, F: father in middle
Solution: We assume that $\mathrm{M}$ denote mother, $\mathrm{F}$ denote father and $\mathrm{S}$ denote son. Then, the sample space for the given experiment will be: S= ${$MFS, SFM, FSM, MSF, SMF,...
A die is thrown three times, E: 4 appears on the third toss,
and 5 appears respectively on first two tosses.
In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $1 \leq x, y, z \leq 6$. Considering the event, E: 4...
A die is thrown three times, E: 4 appears on the third toss,
and 5 appears respectively on first two tosses.
Solution: In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $\1 leq x, y, z leq 6$. Considering the event,...
A die is thrown three times, E: 4 appears on the third toss,
and 5 appears respectively on first two tosses.
Solution: In the sample space, there are 216 outcomes, with each element of the sample space having three entries and taking the form $(x, y, z)$ where $1 \leq x, y, z \leq 6$. Considering the...
Two coins are tossed once, where (i) E: tail appears on one coin,
: one coin shows head
(ii) E: no tail appears, F: no head appears
Solution: Determining the sample space of the given experiment is $\mathrm{S}=\{\mathrm{HH}, \mathrm{HT}, \mathrm{TH}, \Pi\}\}$ (i) Given that, E: tail appears on one coin And F: one coin shows head...
Determine P (E|F) in: A coin is tossed three times, where (i)
: head on third toss,
: heads on first two tosses (ii)
: at least two heads,
: at most two heads (iii)
: at most two tails,
at least one tail
Solution: The sample space of the given experiment will be: S={HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (i) Given that, E: head-on third toss And F: heads on first two tossesSo writing the sample...
If
and
, find
(i) ![Rendered by QuickLaTeX.com P(A \cap B)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9e162f356a6a6942d502389cce0160ed_l3.png)
(ii) ![Rendered by QuickLaTeX.com P(A \mid B)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-45e4d30f3c84db00282f092ca173ad5a_l3.png)
(iii) ![Rendered by QuickLaTeX.com P(B \mid A)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d00910c23a9cdd28c571feae6e730a68_l3.png)
Solution: Given: $\mathrm{P}(\mathrm{A})=\frac{6}{11}, \mathrm{P}(\mathrm{B})=\frac{5}{11}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{7}{11}$ (i) We know that $\mathrm{P}\left(\mathrm{A}^{*}...
Evaluate
, if
and
.
Solution: i) Given relation $2 \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{B})=\frac{5}{13}$ and $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{2}{5}$ $\Rightarrow \mathrm{P}(\mathrm{B})=\frac{5}{13},...
If
and
, find
(i) ![Rendered by QuickLaTeX.com P(A \cap B)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9e162f356a6a6942d502389cce0160ed_l3.png)
(ii) ![Rendered by QuickLaTeX.com P(A \mid B)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-45e4d30f3c84db00282f092ca173ad5a_l3.png)
(iii) P (A U B)
Solution:Given $\mathrm{P}(\mathrm{A})=0.8, \mathrm{P}(\mathrm{B})=0.5$ and $\mathrm{P}(\mathrm{B} \mid \mathrm{A})=0.4$ (i) By definition of conditional probability, we know that,...
Compute
, if
and ![Rendered by QuickLaTeX.com P(A \cap B)=0.32](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c95eab9ea04199d309247b149bff0281_l3.png)
Solution: Given: $P(B)=0.5$ and $P(A \cap B)=0.32$ We know this because conditional probability is defined as follows: $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap...
Given that
and
are events such that
and
, find
and ![Rendered by QuickLaTeX.com P(F \mid E)](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-c42ce8a4d8ec7124edc83c408b1bf3ed_l3.png)
Solution: Given: $P(E)=0.6, P(F)=0.3$ and $P(E \cap F)=0.2$ We know this because conditional probability is defined as follows: $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$ By substituting the values we...