Question

a bag contains 15 red counters and 12 white counters. 2 counters are taken out in succession. their shades are recorded and neither is replaced. find the probability of getting different shades.

Explanation:

Step1: Calculate total counters

Total counters = Red + White = \( 15 + 12 = 27 \)

Step2: Probability of red then white

Probability of red first: \( \frac{15}{27} \)
After removing 1 red, total counters = \( 26 \), white counters = \( 12 \)
Probability of white next: \( \frac{12}{26} \)
Probability (red then white) = \( \frac{15}{27} \times \frac{12}{26} = \frac{15 \times 12}{27 \times 26} = \frac{180}{702} \)

Step3: Probability of white then red

Probability of white first: \( \frac{12}{27} \)
After removing 1 white, total counters = \( 26 \), red counters = \( 15 \)
Probability of red next: \( \frac{15}{26} \)
Probability (white then red) = \( \frac{12}{27} \times \frac{15}{26} = \frac{12 \times 15}{27 \times 26} = \frac{180}{702} \)

Step4: Total probability of different shades

Add the two probabilities: \( \frac{180}{702} + \frac{180}{702} = \frac{360}{702} = \frac{20}{39} \)

Answer:

\(\frac{20}{39}\)