Lessons 33 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
33
Random events and the idea of probability
Textbook, Part 2: pp. 68–77
GoalLearn to tell random, certain and impossible events apart and to use the classical and statistical definitions of probability.
An experiment is a trial that can be repeated many times under unchanged conditions (tossing a coin or a die, drawing a ball from a bag). An event whose result is not known in advance, which may or may not occur, is random; one that occurs every time is certain (U) and one that never occurs is impossible (V). If the outcomes are equally likely, with n in total and k of them favourable to event A, the classical definition gives P(A) = k/n. So P(U) = 1, P(V) = 0 and 0 ≤ P(A) ≤ 1 for every event. Statistical definition: if in n repetitions event A occurs μ times, the ratio μ/n is the frequency; for large n it “oscillates” around some number p, and we take p as the probability. Probability does not mean “exactly this many times in a hundred trials”, but the average share over a long series.
Worked examples
A bag holds 12 balls: 5 red, 4 blue, 3 green. One is drawn at random. P(red) = 5/12, P(blue or green) = (4 + 3)/12 = 7/12. P(yellow) = 0, since it is an impossible event.
When two dice are tossed there are 6 · 6 = 36 outcomes. The pairs with sum 8 are (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) — 5 pairs, so P = 5/36. A drawing pin was tossed 200 times and landed point up 126 times; the frequency 126/200 = 0.63 is a statistical estimate of a probability that cannot be found classically.
Class activity
“Coin experiment”: toss a coin 50 times and after every 10 tosses record the frequency of heads. Do the frequencies fluctuate around 0.5? Pool the results of the whole class and observe how the frequency changes as n grows.
Practice
1
A card is drawn at random from cards numbered 1 to 20. What is the probability that its number is divisible by 5?
1/5
2
What is the probability of a prime number when a die is tossed?
1/2
3
12 of 30 students are girls. What is the probability that a randomly chosen student is a girl?
0.4
4
Why does heads not have to appear exactly 50 times in 100 tosses?
A probability of 1/2 means that the share of heads approaches 1/2 in a long series, not that exactly half appear in each short series. Each toss is random and independent of the previous ones.