A bag contains 4 white and 6 black balls. If two balls are drawn at random, what

A bag contains 4 white and 6 black balls. If two balls are drawn at random, what is the probability that both are black?

A bag contains 4 white and 6 black balls. If two balls are drawn at random, what is the probability that both are black?

Step 1: Count the total number of balls in the bag. There are 4 white balls and 6 black balls, so total = 4 + 6 = 10 balls.
Step 2: Determine the number of black balls. There are 6 black balls.
Step 3: When drawing the first ball, the probability of it being black is the number of black balls divided by the total number of balls: P(first black) = 6/10.
Step 4: If the first ball drawn is black, there will be 5 black balls left and the total number of balls will be 9 (since one ball has been removed).
Step 5: Now, calculate the probability of drawing a second black ball: P(second black | first black) = 5/9.
Step 6: To find the probability of both events happening (drawing two black balls), multiply the probabilities from Step 3 and Step 5: P(both black) = (6/10) * (5/9).
Step 7: Simplify the multiplication: (6/10) * (5/9) = 30/90.
Step 8: Simplify 30/90 to get the final probability: 30/90 = 1/3.

Probability of Independent Events – The question tests the understanding of calculating the probability of drawing two specific outcomes (both black balls) from a finite sample space.
Combinatorial Probability – It involves understanding how to calculate probabilities when drawing without replacement, affecting the total number of outcomes.