Lesson 6 of 11 · 10 min
Probability of an event
NCERT §14.2.1
The mela's trick die is weighted: the six turns up five times as often as each other face. The stall offers a prize for an even number. With a fair die that would be a half. Is it still?
The lesson in notes
In short
Once each outcome has its probability, P(A) is simply the sum of the probabilities of the outcomes inside A.
Three consecutive pens from a machine, each good (G) or defective (B): S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG}. With 1/8 on each outcome, 'exactly one defective' = {BGG, GBG, GGB} has P = 3/8 and 'at least two defective' has P = 4/8 = 1/2.
Outcomes need not carry equal weight. For two tosses NCERT uses P(HH) = 1/4, P(HT) = 1/7, P(TH) = 2/7, P(TT) = 9/28, which add to 1.
Then 'both tosses give the same result' = {HH, TT} has P = 1/4 + 9/28 = 4/7, and 'exactly two heads' = {HH} has P = 1/4.
Mela trick die (made up): face 6 is weighted to be w times as likely as each other face. The weights add to w + 5, so P(6) = w/(w + 5) and every other face gets 1/(w + 5).
With w = 5: P(6) = 1/2 and each other face 1/10. So P(even) = 1/10 + 1/10 + 1/2 = 7/10, P(greater than 4) = 1/10 + 1/2 = 3/5 and P(prime) = 3/10. With w = 1 the die is fair and P(6) = 1/6.