Lessons · Teaching · probability and counting, without a formula sheet
Probability, counted rather than felt
A probability is the number of outcomes you want over the number of outcomes there are, when the outcomes are equally likely. Independent events multiply. Either-or events add, as long as they cannot both happen. Everything on this test comes from those three sentences.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
Probability is where intuition is worst and arithmetic is easiest, which is an unusual and useful combination. It is also the topic children find genuinely exciting, so knowing it properly is worth more in a classroom than its weight on the test.
How to think about it
Count the total outcomes first and write the number down. Then count the ones that satisfy the question. Decide whether events are independent, which means one does not change the other, and multiply if they are. Sanity-check that the answer is between 0 and 1.
Worked example
One die: probability of a number above 4 is 2 out of 6, which is 1/3Two faces out of six. Count, do not estimate.
Two coins both heads: 1/2 x 1/2 = 1/4Independent events multiply. Four equally likely outcomes and one of them is what you want.
Drawing two reds from a bag without replacing the first changes the second probability3 red of 10, then 2 red of 9: 3/10 x 2/9 = 6/90 which is 1/15.
The probability of something NOT happening is 1 minus the probability that it doesOften much less work than counting the case you were asked about.
Your turn
Name the operation that combines two independent events both happening.
Independent events both happening: you the probabilities
Solve one, graded on the server
The trap
The gambler's belief that a run changes the next outcome. A coin that has landed heads five times is still even money, and every version of this question on the test is checking exactly that.