Probability and counting

Probability: the ways you want over all the ways

How to solve them

1. 1

   Count the ways you want, then count all the ways

   Two counts, then one division. First count the favorable outcomes: the ones the question asks about. Then count the total: every outcome, the favorable ones included. The total is the whole group, never just “the others”.

   A die (plural: dice) is a cube with the numbers 1 to 6 on its faces. A standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades) of 13 cards each. Each suit has an ace, the numbers 2 to 10, a jack, a queen and a king, so there are 4 of each value: 4 aces, 4 kings, and so on. Hearts and diamonds are red; clubs and spades are black.

   P(green) is short for “the probability of green”. So P(green) = green ÷ all marbles.

   1. P(green) = green ÷ all marbles
   2. 6 ÷ (6 + 9)
   3. 6 ÷ 15
   4. 2/5
   5. 0.4 = 40%

   One die

   Probability of rolling a 6

   1 favorable face out of 6: 1/6

   A deck

   Probability of drawing a red card

   26 red cards out of 52: 26/52 = 1/2

   A jar

   Probability of drawing a yellow marble from the jar

   9 out of 15: 9/15 = 3/5

   Shortcut

   Simplify at the end

   Answer choices are usually fractions in lowest terms. Write favorable over total first, then divide the top and bottom by the same number: 26/52 becomes 1/2 when you divide both by 26.

   Next step: Not ↓
2. 2

   The chance it does NOT happen: 1 minus

   P(not A) = 1 − P(A)

   Something either happens or it does not, so the two probabilities add up to 1, or 100%. This “does not happen” event is called the complement. When the question says not, find the easy probability and subtract it from 1.

   A forecast says there is a 20% chance of rain. The chance of no rain is 100% − 20% = 80%.

   Example

   A spinner has 8 equal parts, and 3 are red. What is the probability of NOT landing on red?

   1. The probability of red is 3 out of 8: 3/8.
   2. Not red is everything else: 1 − 3/8.
   3. Write 1 as 8/8, so 8/8 − 3/8 = 5/8.
   4. Check by counting: 8 − 3 = 5 parts are not red, out of 8.

   Answer 5/8

   1. P(not red)
   2. 1 − 3/8
   3. 8/8 − 3/8
   4. 5/8

   Next step: And: multiply ↓
3. 3

   Two independent events: multiply

   Two events are independent when one does not change the chances of the other. A coin and a die are independent: the coin has no idea what the die did. For the probability that both happen, multiply.

   P(A and B) = P(A) × P(B)

   A coin and a die: all the outcomes

   	1	2	3	4	5	6
   Heads	H1	H2	H3	H4	H5	H6
   Tails	T1	T2	T3	T4	T5	T6

   There are 2 × 6 = 12 outcomes, and only one of them is heads and a 6. That is why multiplying works.

   1. P(heads and 6)
   2. 1/2 × 1/6
   3. 1/12

   Example

   Each day, one name is picked at random from the same 8 names for kitchen duty. What is the probability that Diaz is picked on both Monday and Tuesday?

   1. Monday: 1 name out of 8, so 1/8.
   2. Tuesday is a new pick from the same 8 names: 1/8 again.
   3. Both days: 1/8 × 1/8 = 1/64. “And” makes the chance smaller.

   Answer 1/64

   Next step: Put back or not ↓
4. 4

   With and without replacement

   When you draw twice from the same bag, ask: does the first item go back in? With replacement means you put it back, so the second draw looks exactly like the first. Without replacement means you keep it, so the second draw has one fewer item in all, and one fewer of the kind you took.

   Words that mean without replacement: “does not put it back”, “keeps the first one”, “two at once”, or picking two different people. “One after the other” alone does not tell you: look for whether the first item goes back.

   1. P(lemon, then lemon), put back
   2. 4/10 × 4/10
   3. 16/100
   4. 4/25

   1. P(lemon, then lemon), not put back
   2. 4/10 × 3/9
   3. 12/90
   4. 2/15

   Next step: Counting ↓
5. 5

   Counting choices: multiply the options

   The multiplication principle: if you make one choice and then another, the number of ways is the number of options for the first times the number of options for the second. With three choices, multiply all three.

   Think of slots. Draw one blank for each choice, write how many options fit in each blank, and multiply. For codes, ask if a digit or letter can repeat. There are 10 digits (0 to 9) and 26 letters (A to Z).

   1. 3 shirts × 4 pants × 3 caps
   2. 12 × 3
   3. 36 outfits

   A PIN

   4 digits, repeats allowed

   10 × 10 × 10 × 10 = 10,000

   Letter, then digit

   A code like B7

   26 × 10 = 260

   Meal deal

   5 mains and 4 drinks

   5 × 4 = 20 meals

   Shortcut

   “And then” means multiply

   Adding the options (3 + 4 + 3 = 10) counts single items, not outfits. If each choice comes with every other choice, multiply.

   Next step: Arrangements ↓
6. 6

   Arrangements: the choices shrink by one

   An arrangement (also called a permutation) is a way to put things in order. The slot idea still works, with one change: once a person or object is used, it cannot be used again. So each slot has one option fewer than the slot before.

   Five people stand in a line for a photo. Any of the 5 can stand first. Then 4 are left for second place, 3 for third, 2 for fourth, and the last person takes the last spot.

   1. 5 × 4 × 3 × 2 × 1
   2. 20 × 3 × 2 × 1
   3. 60 × 2 × 1
   4. 120 ways

   First and second place

   8 runners in a race

   8 × 7 = 56 ways

   Three letters

   Ways to order the letters of CAT

   3 × 2 × 1 = 6: CAT, CTA, ACT, ATC, TCA, TAC

   Shortcut

   The factorial sign

   5! (read “5 factorial”) means 5 × 4 × 3 × 2 × 1 = 120. When the question only fills some places, stop early: first and second place out of 8 is just 8 × 7.

   Next step: Three examples ↓
7. 7

   Three worked examples, from easy to test level

   Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

   The test-level one draws two names without putting the first back. Change the fraction for the second draw before you multiply: one fewer of the kind you took, and one fewer in all.

   Example · Easy

   A box has 5 chocolate bars, 9 granola bars and 6 protein bars. You grab one without looking. What is the probability it is a granola bar?

   First try9 granola bars against 11 others: 9/11

   1. 9/11 compares granola with the other bars. The bottom of the fraction must be all the bars, granola included.
   2. Total: 5 + 9 + 6 = 20 bars.
   3. Favorable: the 9 granola bars.
   4. Probability: 9/20, which is 45/100 = 0.45 = 45%.

   Answer 9/20 (45%)

   Example · Medium

   A base gate uses a 4-digit code. The first digit cannot be 0, and digits can repeat. How many codes are possible?

   First try9 + 10 + 10 + 10 = 39

   1. Adding counts the digits you could press, not the codes. Every first digit goes with every second digit, so multiply.
   2. Four slots. The first has 9 options (1 to 9). Each of the other three has 10 options (0 to 9).
   3. 9 × 10 × 10 × 10: the three 10s make 1,000.
   4. 9 × 1,000 = 9,000.

   Answer 9,000 codes

   Example · Test level

   A hat holds the names of 3 sergeants and 5 privates. Two names are drawn, one after the other, and the first name is not put back. What is the probability that both are sergeants?

   First try3/8 × 3/8 = 9/64

   1. The first try uses 3/8 twice, as if the first name went back in the hat. It did not.
   2. First draw: 3 sergeants out of 3 + 5 = 8 names, so 3/8.
   3. If a sergeant came out, 2 sergeants are left, out of 7 names: 2/7.
   4. Both: 3/8 × 2/7 = 6/56. Divide the top and bottom by 2: 3/28.

   Answer 3/28

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.