Introduction to the Problem

Understanding the Setup

A bag contains 10 white and 6 black balls. We need to calculate the probability of drawing 4 balls successively in such a way that they are alternatively of different colors. This means the sequences must alternate between white and black, specifically WBWB or BWBW.

Total Balls and Drawing Sequences

The total number of balls in the bag is 16 (10 white 6 black). The two sequences that satisfy the condition of alternating colors are:

White Black White Black (WBWB) Black White Black White (BWBW)

Calculating the Probability for Each Sequence

The calculation of the probability for each sequence involves determining the individual probabilities of drawing the balls in the specified order and then multiplying these probabilities together.

Probability of Drawing in the Sequence WBWB

First Ball: White

Probability of drawing a white ball: PW frac{10}{16}

Second Ball: Black

Probability of drawing a black ball after one white is drawn: PB frac{6}{15}

Third Ball: White

Probability of drawing a white ball after drawing one white and one black: PW frac{9}{14}

Fourth Ball: Black

Probability of drawing a black ball after drawing one white and one black: PB frac{5}{13}

The total probability for the sequence WBWB is:

PWBWB frac{10}{16} times frac{6}{15} times frac{9}{14} times frac{5}{13}

Calculating this Probability

PWBWB frac{10 cdot 6 cdot 9 cdot 5}{16 cdot 15 cdot 14 cdot 13}

frac{2700}{43680} approx 0.0619

Probability of Drawing in the Sequence BWBW

First Ball: Black

Probability of drawing a black ball: PB frac{6}{16}

Second Ball: White

Probability of drawing a white ball after one black is drawn: PW frac{10}{15}

Third Ball: Black

Probability of drawing a black ball after drawing one black and one white: PB frac{5}{14}

Fourth Ball: White

Probability of drawing a white ball after drawing one black and one white: PW frac{9}{13}

The total probability for the sequence BWBW is:

PBWBW frac{6}{16} times frac{10}{15} times frac{5}{14} times frac{9}{13}

Calculating this Probability

PBWBW frac{6 cdot 10 cdot 5 cdot 9}{16 cdot 15 cdot 14 cdot 13}

frac{2700}{43680} approx 0.0619

Total Probability of Drawing 4 Balls Alternating in Color

The total probability of drawing 4 balls alternatively of different colors is the sum of the probabilities of the two sequences:

Palternating PWBWB PBWBW frac{2700}{43680} frac{2700}{43680} frac{5400}{43680}

frac{1}{8.08} approx 0.1234

Conclusion and Final Answer

The probability that 4 balls drawn successively from the bag are alternatively of different colors is approximately 0.1234 or 12.34%.