Probability flashcards

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What is the range of values for a probability, and what do 0 and 1 mean?
Between 0 and 1 inclusive. 0 means the event is impossible and 1 means it is certain. Probabilities can be fractions, decimals or percentages.
How do you calculate the probability of an event when outcomes are equally likely?
\(P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\).
What is \(P(A')\) in terms of \(P(A)\)?
\(P(A')=1-P(A)\), because an event either happens or does not, and the probabilities sum to 1.
A bag has 5 red, 3 blue and 2 green counters. Find the probability of picking a blue counter.
\(\frac{3}{10}=0.3\).
What is relative frequency and how is it used to estimate a probability?
Relative frequency \(=\frac{\text{number of times the event happens}}{\text{number of trials}}\). The more trials, the better it estimates the true probability.
A biased spinner lands on 4 in 18 of 60 spins. Estimate \(P(4)\) and the expected number of 4s in 200 spins.
\(P(4)\approx\frac{18}{60}=0.3\). Expected count \(=0.3\times200=60\).
How do you find the expected number of times an event occurs?
Multiply the probability by the number of trials: expected frequency \(=P(\text{event})\times n\).
When two events \(A\) and \(B\) are mutually exclusive, what is \(P(A\text{ or }B)\)?
\(P(A\text{ or }B)=P(A)+P(B)\). Mutually exclusive means they cannot happen together.
When two events are independent, what is \(P(A\text{ and }B)\)?
\(P(A\text{ and }B)=P(A)\times P(B)\). Independent events do not affect each other, e.g. successive coin tosses.
A fair coin is tossed and a fair die is rolled. Find P(head and a 6).
\(\frac12\times\frac16=\frac1{12}\).
What do the branches of a tree diagram show, and how do you use them?
Each branch shows an outcome with its probability. Multiply along branches for the probability of a combined outcome and add the results of different paths for 'or'.
A bag has 4 red and 6 blue counters. Two are taken with replacement. Find P(both red).
\(\frac4{10}\times\frac4{10}=0.16\).
The same bag: two counters are taken without replacement. Find P(both red).
\(\frac4{10}\times\frac39=\frac{12}{90}=\frac2{15}\). After the first is removed, 3 reds remain among 9 counters.
What is the difference between 'with replacement' and 'without replacement' for probabilities?
With replacement, the probabilities stay the same each pick (independent). Without replacement, the total and possibly the favourable number drop, so later probabilities change.
Using the bag of 4 red and 6 blue counters, find P(two counters of different colours) without replacement.
\(\frac4{10}\times\frac69+\frac6{10}\times\frac49=\frac{24+24}{90}=\frac{8}{15}\).
How do you list all outcomes of two events systematically?
Use a sample-space (two-way) table or an organised list. E.g. for two dice, 36 outcomes; \(P(\text{total }7)=\frac6{36}=\frac16\).
Two fair dice are rolled. Find P(total of at least 10).
Totals 10, 11, 12 occur in \(3+2+1=6\) ways. So \(\frac6{36}=\frac16\).
What do the symbols \(\cup\), \(\cap\), \(A'\), \(\varnothing\) and \(\in\) mean in set notation?
\(\cup\): union (in A or B or both). \(\cap\): intersection (in both). \(A'\): complement (not in A). \(\varnothing\): empty set. \(\in\): is an element of.
What does \(n(A)\) mean and what is \(\mathscr{E}\)?
\(n(A)\) is the number of elements in set \(A\). \(\mathscr{E}\) is the universal set containing all elements being considered.
In a Venn diagram, how do you fill in the regions when given \(n(A\cap B)\)?
Start with the intersection, then subtract it from \(n(A)\) and \(n(B)\) to get the 'only A' and 'only B' regions. Put what is left of the total outside both circles.
In a class of 30, 18 study French, 14 study Spanish and 5 study neither. How many study both?
30 \(-\) 5 \(=\) 25 study at least one. \(18+14-25=7\) study both.
Using a Venn diagram, how do you find \(P(A\cup B)\)?
\(\frac{n(A\cup B)}{n(\mathscr{E})}\), the count of elements in either circle (counting the overlap once) over the total in the universal set.
(Extended) State the conditional probability notation and what \(P(A\mid B)\) means.
\(P(A\mid B)\) is the probability of \(A\) given that \(B\) has already happened. Restrict attention to outcomes in \(B\): \(P(A\mid B)=\frac{n(A\cap B)}{n(B)}\).
(Extended) In a Venn diagram, 12 students are in \(A\cap B\) and 20 are in \(B\). Find \(P(A\mid B)\).
\(P(A\mid B)=\frac{12}{20}=\frac35\).
(Extended) A bag has 3 red and 5 blue counters. Two are taken without replacement. Find P(second is red | first is red).
After a red is removed there are 2 red among 7: \(\frac27\).
(Extended) Using the tree for the bag with 3 red and 5 blue taken twice without replacement, find P(at least one red).
\(1-P(\text{both blue})=1-\frac58\times\frac47=1-\frac{20}{56}=\frac{36}{56}=\frac9{14}\).
Why is 'at least one' often easier to solve using complements?
The opposite, 'none', is usually a single path. So \(P(\text{at least one})=1-P(\text{none})\), saving you from adding many cases.
The probability it rains on a given day is 0.2. Find P(no rain on two independent days).
\(0.8\times0.8=0.64\).

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