Flashcards · 10 cards

Probability & distributions

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Rule
$P(A \cup B) = \,?$
Answer
$P(A) + P(B) - P(A \cap B)$ : add, then take off the overlap counted twice. For mutually exclusive events there is no overlap.
A or BAdd, then take off the overlap counted twice
From the lesson: Probability rules
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What makes two events independent?
Answer
$P(A \cap B) = P(A) \times P(B)$ : one happening doesn't change the chance of the other.
From the lesson: Probability rules
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Three people each try once. What is P(none succeed)?
Answer
Multiply the three failure probabilities. It is not $1 - P(\text{all})$ , which means "not all".
From the lesson: Probability rules
Know it
Three people each try once. What is P(exactly one succeeds)?
Answer
Three routes: only $A$ , only $B$ , only $C$ . Each is one success times two failures; add the three.
From the lesson: Probability rules
Rule
The binomial probability of $r$ successes in $n$ trials?
Answer
$P(X = r) = {}^nC_r\,p^r q^{n - r}$ , with $q = 1 - p$ . Write down what counts as a success first.
A binomial distributionOne bar for each number of successes
From the lesson: The binomial distribution
Rule
The Poisson probability of $r$ events?
Answer
$P(X = r) = \dfrac{e^{-\lambda}\lambda^r}{r!}$ , where $\lambda$ is the mean number of events. For a binomial with large $n$ and small $p$ , use $\lambda = np$ .
The Poisson distributionλ is the mean number of events
From the lesson: The Poisson and normal distributions
Rule
How do you find a probability from a normal distribution?
Answer
Change $x$ to $z = \dfrac{x - \mu}{\sigma}$ and read $\Phi(z)$ , the area to the left, from the table: $P(x_1 < X < x_2) = \Phi(z_2) - \Phi(z_1)$ .
The normal curveChange x to z, then read the area from the table
From the lesson: The Poisson and normal distributions
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You are given the variance. What goes into $z = \dfrac{x - \mu}{\sigma}$ ?
Answer
The standard deviation, the square root of the variance: a variance of 25 means $\sigma = 5$ .
From the lesson: The Poisson and normal distributions
Which method?
WAEC 2019 · Paper 2 · Q12
In an examination, $60\%$ of the candidates passed. If 10 candidates are selected at random, find, correct to four decimal places, the probability that:
at least two of them failed;
exactly half of them passed;
at most two of them failed.
What counts as a success here?
Answer
Count the failures: $F \sim B(10, 0.4)$ . Then "at least two failed" is $1 - [P(F = 0) + P(F = 1)]$ .
From the lesson: The binomial distribution Try the question
Which method?
NECO 2023 · Paper 2 · Q15 (b)
The scores of 5000 students follow a normal distribution with mean 72 and variance 25. Determine the number of students who obtained scores between 80 and 84.
How do you get a number of students?
Answer
Find the probability first, with $\sigma = \sqrt{25} = 5$ : $\Phi(2.4) - \Phi(1.6)$ . Then multiply by 5000.
From the lesson: The Poisson and normal distributions Try the question