Question

1. 80 students each study one science. the table shows some information about these students

a. complete the table
b. what is the probability that the student studies physics?
c. what is the probability that the student is male and does not study biology?
d. what is the probability that the student is female and studies chemistry?
e. what is the probability that the student is not female?
f. what is the probability that the student does not study biology?

Explanation:

Step1: Complete the table

• Biology total:

Since total students \(n = 80\), and from the Physics and Chemistry columns. For Biology, using the formula for the total of a row/column.
We know that \(n(\text{Biology total})=80-(21 + 33)=26\)

• Female - Chemistry:

Since \(n(\text{Chemistry total}) = 21\), and \(n(\text{Male - Chemistry})=21 - n(\text{Female - Chemistry})\). Also, \(n(\text{Female total})=47\), \(n(\text{Female - Biology}) = 18\), \(n(\text{Female - Physics})=33 - 19=14\). Then \(n(\text{Female - Chemistry})=47-(18 + 14)=15\)

• Male - Biology:

Since \(n(\text{Biology total}) = 26\), \(n(\text{Male - Biology})=26 - 18 = 8\)

• Male - Chemistry:

Since \(n(\text{Chemistry total})=21\), \(n(\text{Male - Chemistry})=21 - 15 = 6\)

• Male total:

\(n(\text{Male total})=80 - 47=33\)

Step2: Probability formula

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)=80\) (total number of students)

• b. Probability that a student studies Physics

\(n(\text{Physics}) = 33\), so \(P(\text{Physics})=\frac{33}{80}\)

• c. Probability that a student is male and does not study Biology

\(n(\text{Male and not Biology})=n(\text{Male - Chemistry})+n(\text{Male - Physics})=6 + 19=25\), so \(P=\frac{25}{80}=\frac{5}{16}\)

• d. Probability that a student is female and studies Chemistry

\(n(\text{Female - Chemistry}) = 15\), so \(P=\frac{15}{80}=\frac{3}{16}\)

• e. Probability that a student is not female

\(n(\text{Not female})=n(\text{Male})=33\), so \(P=\frac{33}{80}\)

• f. Probability that a student does not study Biology

\(n(\text{Not Biology})=80 - 26=54\), so \(P=\frac{54}{80}=\frac{27}{40}\)

Answer:

• a. Completed table:

	Biology	Chemistry	Physics	Total
Male	8	6	19	33
Total	26	21	33	80

• b. \(\frac{33}{80}\)
• c. \(\frac{5}{16}\)
• d. \(\frac{3}{16}\)
• e. \(\frac{33}{80}\)
• f. \(\frac{27}{40}\)