Tag: statistics and probability

Questions Related to statistics and probability

Assume that the birth of a boy or girl to a couple to be equally likely,mutually exclusive exhaustive and independent of the other children in the family for a couple having $6$ children the probability that their 'three oldest are boy'is

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $\dfrac{{20}}{{64}}$

  2. $\dfrac{{1}}{{64}}$

  3. $\dfrac{2}{{64}}$

  4. $\dfrac{8}{{64}}$

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Correct Option: C
Explanation:
Given that the birth of a boy or girl to a couple to be equally likely, 
$(1)$ mutually exclusive
$(2)$ exhaustive
$(3)$ independent
$\Rightarrow$ One event does not affect the other 
$P(E)=P(B).P(B).P(B), P(B\ or\ G). P(B\ or\ G)$
$P(B)=$ probability of boy is $\dfrac{1}{2}$
$P(G)=$ probability of girl is $\dfrac{1}{2}$
$P(E)=\dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times 2$ either boy or girls
$=\dfrac{2}{64}$

If  $A$ and $B$ are two independent events such that $P\left({A}^{\prime}\right)=0.7,  P\left({B}^{\prime}\right)=p$ and $P\left( A\cup B \right) =0.8,$ then the value of $p$ is 

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $1$

  2. $0.2$

  3. $0.7$

  4. $0.3$


Show answer
Correct Option: D
Explanation:
$P(A')=0.7\Rightarrow P(A)=1-0.7=0.3$

$P(B)=1-p$ 

$P(A\cup B)=0.8$

$P(A\cap B)=P(A)\times P(B)=0.3(1-p)$ as they are independent events

$\therefore P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$0.8=0.3+1-p-0.3(1-p)$

$0.8=0.3+1-p-0.3+0.3p$

$0.8=1-0.7p\Rightarrow 0.7p=0.2$

$p=\dfrac 27\approx0.3$

One card is drawn from a pack of $52$ cards. The probability that the card picked is either a spade or a king.

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $ \displaystyle \frac{1}{26} $

  2. $ \displaystyle \frac{3}{26} $

  3. $ \displaystyle \frac{4}{13} $

  4. $ \displaystyle \frac{1}{9} $

Show answer
Correct Option: C
Explanation:
Let $A:$ event of drawing a king
$\displaystyle \Rightarrow P(A)=\dfrac{4}{52}=\dfrac{1}{13}$
$B:$ event of drawing a spade $\displaystyle =\dfrac{13}{52}=\dfrac{1}{4}$
$\Rightarrow \displaystyle P\left( A\cap B \right) =\dfrac { 1 }{ 52 } $
$\displaystyle \therefore P\left( A\cup B \right) =P\left( A \right) +P\left( B \right) -P\left( A\cap B \right) =\dfrac { 1 }{ 13 } +\dfrac { 1 }{ 4 } -\dfrac { 1 }{ 52 } =\dfrac { 4 }{ 13 } $

In a survey conducted among 400 students of X standard in Pune district, 187 offered to join Science faculty after X std. and 125 students offered to join Commerce faculty after X, If a student is selected at random from this group. Find the probability that student prefers Science or Commerce faculty.

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $\displaystyle \frac{39}{50}$

  2. $\displaystyle \frac{4}{5}$

  3. $\displaystyle \frac{41}{50}$

  4. $\displaystyle \frac{43}{50}$

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Correct Option: A
Explanation:

Total number of students $= 400$
No. of science students $= 187$
No. of commerce students $ = 125$
Thus, Probabilty of the student being either science or commerce = $\dfrac{187}{400} + \dfrac{125}{400}$
= $\dfrac{312}{400}$
= $\dfrac{39}{50}$

An integer is chosen at random from the first two hundred digits.What is the probability that the integer chosen is divisible by 6 or 8 ?

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $\displaystyle\frac{29}{100}.$

  2. $\displaystyle\frac{1}{4}.$

  3. $\displaystyle\frac{1}{8}.$

  4. $\displaystyle\frac{21}{100}.$

Show answer
Correct Option: B
Explanation:

Let A=the integer is divisible by 6 
$A={6,12,18,....198}$
$198=6+(n-1)6$
$\Rightarrow n =33$
So, $P(A)=\dfrac{33}{200}$

B=the integer is divisible by 8
$B={8,16,24,....200}$
$200=8+(n-1)8$
$\Rightarrow n=25$
So, $P(B)=\dfrac{25}{200}$

Both are divisible by 6 and 8 both 
$A\cap B=24,48,.....192$
$192=24+(n-1)24$
$\Rightarrow n=8$
So, $P(A\cap B)=\dfrac{8}{200}$

$\displaystyle \therefore P\left ( A\cup B \right )=P\left ( A \right )+P\left ( B \right )-P\left ( AB \right )$
$\displaystyle=\frac{33}{200}+\frac{25}{200}-\frac{8}{200}=\frac{1}{4}.$

A random variable $X$ has the probability distribution:

$x$ 1 2 3 4 5 6 7 8
$P(X=x)$ 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events $E=\left {x|x \text {is prime}\right }$ and $F=\left {x|x < 4\right }$, the probability $P(E\cup F)$ is

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $0.35$

  2. $0.77$

  3. $0.87$

  4. $0.50$

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Correct Option: B
Explanation:

Using given table, $P(E)=0.62, P(F)=0.5, P(E\cap F)=0.35$.
$\therefore P(E\cup F)=P(E)+P(F)-P(E\cap F)=0.62+0.5-0.35=.77$

A die is thrown. Let $A$ be the event that the number obtained is greater than $3$. Let $B$ be the event that the number obtained is less than $5$. Then $P(A \cup B)$

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $3/5$

  2. $0$

  3. $1$

  4. $2/5$

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Correct Option: C
Explanation:

Simply speaking, any number on the dice is either less than $5$ or more than $3$ (since the range covers all the numbers from $1$ to $6$). Hence, required probability = $1$.

Alternate method:
$P(A\cup  B)=P(A)+P(B)-P(A\cap B)=\dfrac { 3 }{ 6 } +\dfrac { 4 }{ 6 } -\dfrac { 1 }{ 6 } =1$

A die is thrown. Let A be the event that the number obtained is greater than 3. Let$B$ be the event that the number obtained is less than 5. Then $P(A\cup B)$ is 

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. 1

  2. $\displaystyle \frac{2}{5}$

  3. $\displaystyle \frac{3}{5}$

  4. $0$

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Correct Option: A
Explanation:

$n(A\cup B)={1,2,3,4,5,6}\therefore  P(A\cup B)=1$

A card is drawn is from pack of 52 cards. Find the probability of drawing '5' of spade or '8' of hearts

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $\dfrac{1}{52}$

  2. $\dfrac{1}{26}$

  3. $\dfrac{3}{52}$

  4. $\dfrac{1}{13}$

Show answer
Correct Option: B
Explanation:

$\displaystyle P(A)=\frac { 1 }{ 52 } +\frac { 1 }{ 52 } =\frac { 2 }{ 52 } =\frac { 1 }{ 26 } $

A box contains 5 red balls, 8 green balls and 10 pink balls. A ball is drawn at random from the box. What is the probability that the ball drawn is either red or green?

business maths probability - i addition theorems of probability statistics and probability descriptive statistics and probability

  1. $\dfrac{13}{23}$

  2. $\dfrac{10}{23}$

  3. $\dfrac{11}{23}$

  4. $\dfrac{13}{529}$

Show answer
Correct Option: A
Explanation:

Total Number of Balls $= 5+8+10 = 23$ balls

Probability of getting a red ball  $= \dfrac{5}{23} $

Probability of getting a green ball  $= \dfrac{8}{23} $

Probability of getting a pink ball  $= \dfrac{10}{23} $

Then, probability of getting a red  or a green ball  $= \dfrac{5+8}{23} = \dfrac{13}{23}$