ITPEC IP October 2023 Question 1

Topic: Basic TheoryAlso asked in November 2021

When three balls are picked out consecutively from a box containing three white balls and four black balls, which of the following is the lowest probability?

a)Probability of the balls picked out in order of black, black, blackb)Probability of the balls picked out in order of black, black, whitec)Probability of the balls picked out in order of black, white, blackd)Probability of the balls picked out in order of white, black, black

Explanation

Black, black, black — without replacement, the third black draw has only 2 black balls left out of 5.

The box holds 7 balls, 4 black (B) and 3 white (W). A drawn ball is not returned, so each draw uses what is left:

•B, B, B: 4/7 × 3/6 × 2/5 = 24/210
•B, B, W: 4/7 × 3/6 × 3/5 = 36/210
•B, W, B: 4/7 × 3/6 × 3/5 = 36/210
•W, B, B: 3/7 × 4/6 × 3/5 = 36/210

The denominators are always 7 × 6 × 5 = 210. Every order with two black and one white gives the numerator 4 × 3 × 3 = 36; only B, B, B gives 4 × 3 × 2 = 24.

Answer (a)

Why not others:

•(b), (c) and (d) — each takes two black balls and one white, so all three have the same probability, 36/210, higher than 24/210; the order of the draws changes the factors but not their product

Key rule: Without replacement, reduce both the count of the colour drawn and the total after every draw; sequences with the same colour counts have equal probability whatever their order.

Basic Theory · AI-written, may contain errors

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