Question

Which of the following option resembles $\text{P(A or B)}$ and $\text{P(A and B)}$, respectively?

• A. $\text{P(A)+P(B)}$ and $\frac{\text{P(A)}}{\text{P(B)}}$
• B. $\text{P(A)-P(B)}$ and $\frac{\text{P(A)}}{\text{P(B)}}$
• C. $\text{P(A)+P(B)}$ and $\text{P(A)}\times \text{P(B)}$
• D. $\text{P(A)-P(B)}$ and $\text{P(A)}\times \text{P(B)}$

Answer 1

The correct option is C $\text{P(A)+P(B)}$ and $\text{P(A)}\times \text{P(B)}$

$\text{P(A or B)}$ signifies the probability of occurrence of either $\text{A}$ or $\text{B}$.

$\text{Example:}$ Let's find out the probability of rolling $1$ or $6$ in a six-sided fair die.

By conventional method, we can write
$\text{P(rolling 1 or 6)}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

$\Rightarrow \text{P(rolling 1 or 6)}=\frac{2}{6}=\frac{1}{3}$

Alternatively, we can write
$\text{P(rolling 1 or 6)}=\frac{2}{6}$
$\Rightarrow \text{P(rolling 1 or 6)}=\frac{1}{6}+\frac{1}{6}$
$\Rightarrow \text{P(rolling 1 or 6)}=\text{P(rolling 1)}+\text{P(rolling 6)}$

$\therefore \underset{–}{\text{P(A or B)=P(A)+P(B)}}$
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$\text{P(A and B)}$ signifies the probability of occurrence of both $\text{A}$ and $\text{B}$.

$\text{Example:}$ Tamyra flips a coin and rolls a die simultaneously. Let's find out the probability of flipping tail and rolling $5$.

Sample space for flipping a coin and rolling a die= $\text{{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}}$

By conventional method, we can write
$\text{P(flipping tail and rolling 5)}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

$\Rightarrow \text{P(flipping tail and rolling 5)}=\frac{1}{12}$

Alternatively, we can write
$\text{P(flipping tail and rolling 5)}=\frac{1}{12}$
$\Rightarrow \text{P(flipping tail and rolling 5)}=\frac{1}{2}\times \frac{1}{6}$
$\Rightarrow \text{P(flipping tail and rolling 5)}=\text{P(flipping tail)}\times \text{P(rolling 5)}$

$\therefore \underset{–}{\text{P(A and B)=P(A)}\times \text{P(B)}}$