Sure! Let's go step-by-step to calculate the probability of [tex]\( P(A \cap B) \)[/tex].
1. Define the set of cards: The set of 10 cards is numbered from 1 to 10:
[tex]\[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \][/tex]
2. Event A: Choosing a number less than 7:
The numbers less than 7 are:
[tex]\[ \{1, 2, 3, 4, 5, 6\} \][/tex]
There are 6 numbers in Event A.
3. Event B: Choosing an odd number:
The odd numbers in the set are:
[tex]\[ \{1, 3, 5, 7, 9\} \][/tex]
There are 5 numbers in Event B.
4. Intersection of Event A and Event B ([tex]\( A \cap B \)[/tex]):
We need to find the numbers that are both less than 7 and odd. These numbers are:
[tex]\[ \{1, 3, 5\} \][/tex]
There are 3 numbers in [tex]\( A \cap B \)[/tex].
5. Calculate the total number of possible outcomes:
There are 10 cards, so there are 10 possible outcomes.
6. Calculate the probability of [tex]\( A \cap B \)[/tex]:
The probability [tex]\( P(A \cap B) \)[/tex] is the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[
P(A \cap B) = \frac{\text{Number of outcomes in } A \cap B}{\text{Total number of possible outcomes}} = \frac{3}{10}
\][/tex]
Therefore, the probability [tex]\( P(A \cap B) \)[/tex] is:
[tex]\[ P(A \cap B) = 0.3 \][/tex]
