Question

Records from a dairy farm yielded the following information on the number of male and female calves born at various times of the day.

What is the probability that a randomly selected calf was born in the night or was a female?

a. $\frac{369}{513}$

b. $\frac{485}{513}$

C. $\frac{116}{513}$

d.$\frac{116}{252}$

e ${}^{\frac{116}{233}}$

Short answer

The probability that a randomly selected calf was born in the night or was a female is$\frac{369}{513}$.

Step-by-step solution

Given information

The table is

Time of day

Gender

	Day	evening	night	total
Males	129	15	117	261
Females	118	18	116	252
Total	247	33	233	513

Formulae used:

Probability$=\frac{\text{Fawourable Cases}}{\text{Total Cases}}$

Complement rule:

$P\left(A^c\right)=P\left(\text{not}\mathrm{A}\right)=1-P\left(A\right)$

Addition rule for any two events:

$P\left(A\text{or}\mathrm{B}\right)=P\left(A\right)+P\left(B\right)-P\left(A\text{and}\mathrm{B}\right)$

Definition conditional probability:

$P(B/A)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P\left(A\text{and}B\right)}{P\left(A\right)}$

Explanation for correct option

It has been discovered that 233 of the 513 people were born at night.

$P\left(\text{Born at night}\right)=\frac{\text{Noof favorable outcomes}}{\text{Noof possible outcomes}}=\frac{233}{513}$

It was also discovered that 252 of the 513 people were female.

$P\left(\text{Female}\right)=\frac{\text{Noof favorable outcomes}}{No\text{of possible outcomes}}=\frac{252}{513}$

Finally, we should mention that 116 of the 513 people were female and were born at night.

$\text{P (Female and Born at night})=\frac{No\text{of faworable outcomes}}{No\text{of possible outcomes}}=\frac{116}{513}$

P( Female and Born at night)=P (Female) +P (Born at night) -P

(Female and Born at night)

$$\begin{gathered} =\frac{233}{513}+\frac{252}{513}-\frac{216}{513} \\ =\frac{223-262-116}{513} \\ =\frac{369}{513} \end{gathered}$$

Therefore, the correct option is (a)

Explanation for incorrect option

(b) The probability that a randomly selected calf was born in the night or was a female is not$\frac{485}{513}$

(c) The probability that a randomly selected calf was born in the night or was a female is not$\frac{116}{513}$

(d) The probability that a randomly selected calf was born in the night or was a female is not $\frac{116}{252}$

(e) The probability that a randomly selected calf was born in the night or was a female is not $\frac{116}{233}$