Question

1. from a set of 10 cards numbered 1 to 10, two cards are drawn at random without replacement. what is the probability that:

3a both numbers are even?
probability = \\(\frac{2}{9}\\)
5 you got it!
3b one is even and one is odd?
probability = enter your next step here

Explanation:

Step1: Determine number of even and odd cards

There are 10 cards (1 - 10). Even numbers: 2,4,6,8,10 (5 cards). Odd numbers: 1,3,5,7,9 (5 cards).

Step2: Calculate total ways to draw 2 cards

Using combination formula \( C(n,r)=\frac{n!}{r!(n - r)!} \), total ways \( C(10,2)=\frac{10!}{2!8!}=\frac{10\times9}{2\times1}=45 \).

Step3: Calculate favorable ways (1 even, 1 odd)

Ways to choose 1 even: \( C(5,1)=\frac{5!}{1!4!}=5 \). Ways to choose 1 odd: \( C(5,1)=5 \). Favorable ways: \( 5\times5 = 25 \).

Step4: Calculate probability

Probability \(=\frac{\text{favorable ways}}{\text{total ways}}=\frac{25}{45}=\frac{5}{9} \).

Answer:

\(\frac{5}{9}\)