Question

Let \[A = \left\{ {1,3,5,7,9} \right\}\], \[B = \left\{ {2,4,6,8} \right\}\]. An element (a, b) of their Cartesian product A×B chosen at random, the probability of a + b = 9 is
1) \[\dfrac{1}{4}\]
2) \[\dfrac{1}{5}\]
3) 1
4) 0

Answer 1

Hint: Here, we are given two sets A and B and we need to find the required probability. So, for that first we need to find the Cartesian product of the A and B. We are given that the total of a + b = 9. So, we need to find the favorable outcomes which have a total of 9. And then, we need to find the required probability which is equal to the total number of favorable outcomes divided by the total number of outcomes and thus, we will get the final output.

Complete answer:
Given that,
\[A = \left\{ {1,3,5,7,9} \right\}\]
\[\therefore n\left( A \right) = 5\]
And,
\[B = \left\{ {2,4,6,8} \right\}\]
\[\therefore n\left( B \right) = 4\]
We will find the value of the Cartesian product of A and B as below:
\[
  \therefore A \times B = \{ \left( {1,2} \right),\left( {1,4} \right),\left( {1,6} \right),\left( {1,8} \right),\;\left( {3,2} \right),\left( {3,4} \right),\left( {3,6} \right),\left( {3,8} \right), \\
  \left( {5,2} \right),\left( {5,4} \right),\left( {5,6} \right),\left( {5,8} \right),\;\left( {7,2} \right),\left( {7,4} \right),\left( {7,6} \right),\left( {7,8} \right), \\
  \left( {9,2} \right),\left( {9,4} \right),\left( {9,6} \right),\left( {9,8} \right)\} \\
 \]
Thus,
\[\therefore n(A \times B)\]
\[ = 5 \times 4\]
\[ = 20\]
We are given that, a + b = 9
So, we need to find the set from the above cartesian product of A and B which have a total of 9.
i.e. \[\left( {1,{\text{ }}8} \right) = 1 + 8 = 9\]
\[\left( {3,6} \right) = 3 + 6 = 9\]
\[\left( {5,4} \right) = 5 + 4 = 9\]
\[\left( {7,2} \right) = 7 + 2 = 9\]
Thus, the favourable outcomes are as below:
Let F be the favourable outcomes, then
\[F = \left\{ {\left( {1,8} \right),\left( {3,6} \right),\left( {5,4} \right),\left( {7,2} \right)} \right\}\]
\[\therefore n\left( F \right) = 4\]
Thus, the required probability be
\[ = \dfrac{{Number{\text{ }}of{\text{ }}favorable{\text{ }}outcomes}}{{Total{\text{ }}number{\text{ }}of{\text{ }}outcomes}}\]
\[ = \dfrac{4}{{20}}\]
\[ = \dfrac{1}{5}\]
Hence, for the given A and B sets, the required probability is \[\dfrac{1}{5}\] .

Note:
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes. Sometimes students get mistaken for a favourable outcome with a desirable outcome.