Tag: maths

Questions Related to maths

Which of the following is true regarding law of total probability?

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. It is a fundamental rule relating marginal probabilities to conditional probabilities.

  2. It expresses the total probability of an outcome which can be realized via several distinct events

  3. Both are correct

  4. None of these

Show answer
Correct Option: C
Explanation:
Law of total probability states that,
        If $B _1,B _2,B _3,...$ is a partition of the sample space $S$, then for any event $A$ we have
$P(A)=\sum _iP(A\cap B _i)=\sum _i P(A|B _i)P(B _i)$

That is, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events and hence the name.

Thus option $(C)$ is correct.

I have three bags that each contain $100$ marbles- Bag $1$ has $75$ red and $25$ blue marbles, Bag $2$ has $60$ red and $40$ blue marbles, Bag $3$ has $45$ red and $55$ blue marbles. I choose one of the bags at random and then pick a marble from the chosen bag, also at random. What is the probability that the chosen marble is red?

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. $0.60$

  2. $0.40$

  3. $0.50$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given that bag 1 contains $75$ red and $25$ blue marbles

Given that bag 2 contains $60$ red and $40$ blue marbles
Given that bag 3 contains $45$ red abd $55$ blue marbles
Now the probability of choosing red ball from bag 1 is $ \dfrac{1}{3} \times \dfrac{75}{100}$
Now the probability of choosing red ball from bag 2 is $ \dfrac{1}{3} \times \dfrac{60}{100}$
Now the probability of choosing red ball from bag 3 is $ \dfrac{1}{3} \times \dfrac{45}{100}$
Now the probability of choosing red ball is $ \dfrac{1}{3}\left(\dfrac{75+60+45}{100}\right)=0.6$ 
Hence, option $A$ is correct.

For a random experiment, all possible outcomes are called

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. numerical space.

  2. event space.

  3. sample space.

  4. both b and c.

Show answer
Correct Option: D
Explanation:

We know that,

An outcome is a result of a random experiment. 
The set of all possible outcomes is called the sample space.
The subset of the sample space is called event space.
Thus, for a random experiment, all possible outcomes are called sample space and event space.
Thus option $(D)$ is correct.

Tossing a coin is an example of .........

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. Infinite discrete sample space

  2. Finite sample space

  3. Continuous sample space

  4. None of these

Show answer
Correct Option: B
Explanation:

A coin is tossed.

The possible outcomes are "$\text{head,tail}$"
Thus $ S={H,T}$
We get a finite sample space.
Thus tossing a coin is an example of finite sample space.

The term law of total probability is sometimes taken to mean the ____

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. Law of total expectation

  2. Law of alternatives

  3. Law of variance

  4. None of these

Show answer
Correct Option: B
Explanation:

The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables.

Hence, option B is correct.

The events $E _1, E _2, ........$ represents the partition of the sample space $S$, if they are:

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. pairwise disjoint

  2. exhaustive

  3. have non-zero probabilities

  4. All are correct

Show answer
Correct Option: D
Explanation:

A set of events $E _1 , E _2 ,...$  is said to represent a partition of a sample space $S$, if 


$(a)$  $E _i \cap E _j = \phi, i\neq j; i, j = 1, 2, 3,..., n$  (pairwise disjoint)

$(b)$ $E _i \cup E _2 \cup ... \cup E _n = S$ (exhaustive)

$(c)$ Each $E _i \neq \phi, i.e, P(E _i) > 0$ for all $i = 1, 2, ..., n$ (have non-zero probabilities)

That is the events should be pairwise disjoint, exhaustive and should have non zero probabilities.

Hence, option D is correct.

The experiment is to repeatedly toss a coin until first tail shows up. Identify the type of the sample space.

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. Finite sample space

  2. Continuous sample space

  3. Infinite discrete sample space

  4. None of these

Show answer
Correct Option: C
Explanation:

The experiment is to repeatedly toss a coin until first tail shows up.

A coin is tossed. The possible outcomes are $head-H, tail-T$.
Look into the following table:

 $1^{st}$ toss  $2^{nd}$ toss  $3^{rd}$ toss  $4^{th}$ toss $5^{th}$ toss 
 $T$ $-$ $-$ $-$  $-$
 $H$  $T$  $-$ $-$  $-$
 $H$  $H$  $T$  $-$ $-$
 $H$  $H$  $H$  $T$  $-$
 $H$  $H$  $H$  $H$  $T$

If we get $T$ at first toss, then our experiment ends.
Otherwise second toss. If we get $T$, out experiment ends. If not the process continues till we end up in tail.
Hence the possible outcomes are sequences of $H$ that, if finite, end with a single $T$, and an infinite sequence of $H$.
Therefore, $ S={T,HT,HHT,HHHT,HHHHT,..., {HHHH....}}$
Thus we get a infinite but countable(depends on the number of toss) sample space.
As we shall see elsewhere, this is a remarkable space that contains a not impossible event whose probability is $0$. 
We know that, "a discrete sample space is one with a finite or countably infinite number of possible values."
Hence, it is a infinite discrete sample space.

The experiment is to randomly select a human and measure his or her length. Identify the type of the sample space.

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. Finite sample space

  2. Continuous sample space

  3. Infinite discrete sample space

  4. None of these

Show answer
Correct Option: B
Explanation:

The experiment is to randomly select a human and measure his or her length. 

Depending of how far reaching our means of selection is it is possible to consider a sample space of about $6.6$ billion humans inhabiting the planet Earth. 
In this case, the height of the selected person becomes a random variable. 
However, it is also possible to consider the sample space consisting of all possible values of height measurements of the world population. 
The tallest man ever measured lived in the United States and had a height of $272$ cm $ (8'11'')$. The height of the shortest person is more difficult to determine. Zero is clearly the low bound, but, for a living adult, it may be safely raised to, say, $40 $ cm. 
This suggests a sample space which is a line segment $[40,272]$ in centimeters. 
While at all times the human population is discrete, we may assume that in some height range near the normal average, all possible heights are realized making a continuous classification. 
We know that, "A continuous sample space is one which takes values in one or more intervals."
Thus the sample space is a continuous sample space.

Given a circle of radius $R$, the experiment is to randomly select a chord in that circle. Identify the type of the sample space.

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. Finite sample space

  2. Continuous sample space

  3. Infinite discrete sample space

  4. None of these

Show answer
Correct Option: B
Explanation:

Given a circle of radius $R$, the experiment is to randomly select a chord in that circle.

There are many ways to accomplish such a selection. 

However the sample space is always the same:

$\{AB: A \:and\: B\: are\: points\: on\: a\: given\: circle\}$.

One natural random variable defined on this space is the length of the chord.

Here the length of the chord takes more values depending on the point we choose in the circle.

We know that, "A continuous sample space is one which takes values in one or more intervals."

Therefore, this is a continuous sample space.

Sample space for experiment in which a dice is rolled is

maths probability - iii theorem of total probability bayes theorem probability and probability distribution

  1. $4$

  2. $8$

  3. $12$

  4. None of these

Show answer
Correct Option: D
Explanation:

A dice is rolled.

Thus $S={1,2,3,4,5,6}$
$\Rightarrow n(S)=6$.
That is, sample space for experiment in which a dice is rolled is $6$.
Since $6$ is not listed in the given options we choose D.