Flashcards · 13 cards

Probability

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Rule
Every outcome is equally likely. How do you find the probability of an event?
Answer
$P = \dfrac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}$
Favourable over possible5 green beads out of 12 equally likely beads
From the lesson: Probability and sample spaces
Know it
What values can a probability take?
Answer
From $0$ (impossible) to $1$ (certain): $0 \le P \le 1$ . A probability above 1 or below 0 means a slip in the working.
From 0 to 1P(A) + P(not A) = 1
From the lesson: Probability and sample spaces
Rule
You know $P(A)$ . What is $P(\text{not } A)$ ?
Answer
$P(\text{not } A) = 1 - P(A)$ .
The complementP(A) + P(not A) = 1
From the lesson: Probability and sample spaces
Rule
How do you find a probability from an experiment (relative frequency)?
Answer
$\dfrac{\text{frequency of the outcome}}{\text{total number of trials}}$
Relative frequencyFrequency of the score ÷ total number of throws
From the lesson: Probability and sample spaces
Rule
$A$ and $B$ cannot happen together. What is $P(A \text{ or } B)$ ?
Answer
$P(A) + P(B)$ : mutually exclusive events add.
Mutually exclusiveNo overlap: P(A or B) = P(A) + P(B)
From the lesson: Probability and sample spaces
Rule
$A$ and $B$ can happen together. What is $P(A \text{ or } B)$ ?
Answer
$P(A) + P(B) - P(A \text{ and } B)$ : adding counts the overlap twice, so take it away once.
OverlappingAdding P(A) and P(B) counts the middle twice, so subtract P(A and B) once
From the lesson: Probability and sample spaces
Rule
$A$ and $B$ are independent. What is $P(A \text{ and } B)$ ?
Answer
$P(A) \times P(B)$ : "and" multiplies.
Both A and BP(A and B) = P(A) × P(B): the corner of the square
From the lesson: Combined events and tree diagrams
Rule
What is the quickest way to find $P(\text{at least one})$ ?
Answer
$1 - P(\text{none})$ : everything except "neither".
At least oneEverything except neither
From the lesson: Combined events and tree diagrams
Rule
On a tree diagram, when do you multiply and when do you add?
Answer
Multiply along a route (this, then that). Add the routes that make the event you want (this or that).
Tree diagramMultiply along a route; add the routes you want
From the lesson: Combined events and tree diagrams
Rule
Two balls are drawn one after the other without replacement. What changes for the second draw?
Answer
There is one fewer ball in total, and one fewer of the colour already taken. With 5 red out of 8: $P(\text{red, then red}) = \dfrac58 \times \dfrac47 = \dfrac{5}{14}$ .
Tree diagramMultiply along a route; add the routes you want
From the lesson: Combined events and tree diagrams
Know it
Two fair dice are thrown. How many equally likely outcomes are there?
Answer
$6 \times 6 = 36$ . Draw the 6 by 6 table of outcomes and count the cells you want.
From the lesson: Probability and sample spaces
Which method?
WAEC 2021 · Paper 2 · Q5 (a)
There are 25 boys and 15 girls in a class, all of them equally likely to be chosen for a contest. If two students are chosen one after the other without replacement for the contest, find the probability that: (i) two boys are chosen; (ii) a boy and a girl are chosen; (iii) two boys or two girls are chosen.
What must you watch for?
Answer
Without replacement, the second fraction is out of 39, not 40. "A boy and a girl" has two routes, boy then girl or girl then boy: add them.
From the lesson: Combined events and tree diagrams Try the question
Which method?
WAEC 2014 · Paper 2 · Q5
A building contractor tendered for two independent contracts, $X$ and $Y$ . The probability that he will win contract $X$ is $0.5$ and that he will not win contract $Y$ is $0.3$ . What is the probability that he will win:
both contracts;
exactly one of the contracts;
neither of the contracts?
What do you do first?
Answer
Turn "not win $Y$ " into "win $Y$ ": $1 - 0.3 = 0.7$ . The contracts are independent, so "both" multiplies. "Exactly one" is two routes: $X$ and not $Y$ , or $Y$ and not $X$ .
From the lesson: Combined events and tree diagrams Try the question