Given the table below showing the results of a sample of 1000 people (525 men and 475 women), of which 113 are left-handed
(63 men and 50 women).
[tex]\begin{center}
\begin{tabular}
{|c||c|c|c|}
& Male & Female & Total \\ [1ex]
Left handed & 63 & 50 & 113 \\
Right handed & 462 & 425 & 887 \\ [1ex]
Total & 525 & 475 & 1000
\end{tabular}
\end{center}[/tex]
A person is selected at random from the sample.
a.) The probability that the person is left handed or female is given by the probability that the person is left handed plus the probability that the person is a female minus the probability that the person is both left handed and a female. This is given by
[tex] \frac{113}{1,000} + \frac{475}{1,000} - \frac{50}{1,000} = \frac{538}{1,000} =0.538[/tex]
b.) The probability that the person is left handed and female is given
by the probability that the person is both left handed and a female. This is given by
[tex]\frac{50}{1,000} =0.05[/tex]
c.) The probability that the person is right handed or male is given
by the probability that the person is right handed plus the probability
that the person is a male minus the probability that the person is
both right handed and a male. This is given by
[tex]\frac{887}{1,000} + \frac{525}{1,000} - \frac{462}{1,000} = \frac{950}{1,000} =0.95[/tex]
d.) The probability that the person is not right handed or is a male is given
by the probability that the person is not right handed plus the probability
that the person is a male minus the probability that the person is
both not right handed and is a male.
Note that if a person is not right handed, then they are left handed.
Thus, the probability that the person is not right handed or is a male is given by
[tex]\frac{113}{1,000} + \frac{525}{1,000} - \frac{63}{1,000} = \frac{575}{1,000} =0.575[/tex]
e.) The probability that the person is right handed and is a female is given
by the probability that the person is both right handed and is a female. This is given by
[tex]\frac{425}{1,000} =0.425[/tex]
f.) The probability that the person is male and a female is zero. Notice that a person cannot be a male and a female at the sme time, so the probability that the person is male and a female is mutually exclusive and hence is equal to zero.
g.) Two events are said to be mutually exclusive, if both events cannot happen at the same time, which means that the probability of the two events happening at the smae time is zero.
From the table, it can be seen that there are 425 females that are right handed. Thus the probability of being right handed and being a female is equal to 0.425 and not zero.
Therefore, "being right handed" and "being a female" in NOT mutually exclusive.
