Here’s my logic.

This is a weighted average of probability. The weights are proportional to the probability of each event.

The first event is the probability of either a black ball or a white ball. Four of the five bálls are black and one is white. The black ball has a (4/5) 0.80 probability and the white ball has a (1/5) 0.20 probability of being drawn

IF a black ball (0.80 probability) is drawn, then bin “b” probabilities are in play.

The weights are calculated the same as above. There are four b***s – three are $1 bálls and one is a $7 ball. The probability of drawing a $1 ball is (3/4) 0.75 and the probability of drawing a $7 is (1/4) 0.25

0.75 * $1 = 0.75

0.25 * 7$ = 1.75

Total = 2.50 average per event

$2.50* 0.80 (Black ball Probability) = $2.00

The logic for the white ball event is the same.

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Ugh! CDD strikes again!!

I’m not posting it because I see from Sir Alan’s post that my numbers are wrong. I was using one $500-ball and eight $8-bálls. (I think I added 1 to the 8 instead of the five).

I added Sir Alan’s solution to Naus’ Tortoise and Hair AI formula for blonds, so it now works for red-heads.

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Here's the fully repaired answer:

Black Ball probability 0.80

$1 Ball drawn probability 0.75 Expected Value = 0.80 * 0.75 * 1 = $0.60

$7 Ball drawn probability 0.25 Expected Value = 0.80 *0.25 * 7 = $1.40

Sum $2.00

White Ball probability 0.20

$8 Ball drawn probability (5/6) 0.8333 Expected Value = 0.20 * 0.8333* 8 = $1.33

$500 Ball drawn probability (1/6) 0.1667 Expected Value = 0.20 * 0.1667 * 500 = 16.67

Sum $18.00

18.00+2.00 = $20.00 Expected Value