Question 5 and 6 Exercise 6.5

Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

A student finds that probability of passing an algebra rest is $\dfrac{8}{9}$. What is the probability of failing the test?

Here we can differentiate as: Let $$E=\{ event\, passing\, the\, test \}$$ and $$E^{\prime}=\{ event\, failing\, the\, test \}$$ $E$ and $E^{\prime}$ are complementary events, and we are given $P(E)=\dfrac{8}{9}$, therefore, $$\begin{align}P(E^{\prime})&=1-P(E)=1-\dfrac{8}{9}=\dfrac{1}{9}\end{align}$$

In the two dice experiment, given that the the first dice shows $4$, what is the probability that the second dice shows a number greater than $4$?

The sample space of throwing the second dice is $$\begin{align}S&=\{1,2,3,4,5,6\}\\ n(S)&=6 \end{align}$$ Let $$\begin{align}A&=\{ a\, number\, greater\, tha, 4\}\\ &=\{5,6\}\\ \text{then} n(A)&=2\end{align}$$ Thus, the probability that the second dice shows a number greater than $4$ is: $$\begin{align}P(A)&=\dfrac{n(A)}{n(S)}\\ &=\dfrac{2}{6}\\ &=\dfrac{1}{3}\end{align}$$

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