91Ó°ÊÓ

The table is

Time of day

Gender

Formulae used:

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

Complement rule:

Addition rule for any two events:

Definition conditional probability:

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)

(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 $\frac{116}{233}$