Two bags and have equal number of balls. Bag has 20% red balls and 80% green balls. Bag has 30% red balls, 60% green balls and 10% yellow balls. Contents of Bags and are mixed thoroughly and a ball is randomly picked from the mixture. What is the chance that the ball picked is red?
20%
25%
30%
40%
ANSWER
The probability that the ball picked is red is 25%, option (B).
Official answer: BSource: Official answer key, GATE 2018 IN (IIT Guwahati)
Let each bag hold balls, so the mixture holds balls.
Red balls from bag : . Red balls from bag : . Total red balls in the mixture: .
Every ball in the thoroughly mixed pile is equally likely to be picked, so by classical probability (favourable over total):
Check with the total probability theorem: since the bags contribute equal numbers of balls, a ball drawn from the mixture came from or from with probability each, so
The answer is 25%, option (B).
KEY IDEA
When equal-sized groups are pooled, the proportion of a colour in the mixture is the plain average of the group proportions: . This is the total probability theorem with . The yellow balls in bag make no difference, because only the red fraction of each bag is used.
WATCH OUT FOR
Adding the two percentages () without dividing by the doubled total, which gives an option that is not even listed. Another mistake is letting the green or yellow split change the answer. Note too that the simple average works only because the bags hold equal numbers of balls. With unequal bags the average must be weighted by bag size.
From the GATE 2018 IN paper. Every question of this paper · Download the full paper · All 33 Probability & statistics questions, with notes
