Tree Diagrams & Combined Probability Flashcards

All 12 cards in this deck

What must the probabilities on a set of branches coming from one point on a tree diagram add up to?

$1$.
e.g. if one branch is $0.65$, the other must be $0.35$.

What are the steps to complete a probability tree diagram for two successive independent events?

e.g. Dan's probability of winning each game is $0.3$

1. Complete the first pair: $1-0.3=0.7$ for 'does not win'
2. Write the same probabilities on every second pair: $0.3$ and $0.7$
3. Check each pair of branches sums to $1$

What are the steps to complete the second set of branches of a tree diagram for two picks without replacement?

e.g. a bag has 3 red and 2 blue counters, and the first counter taken is red

1. Reduce the total by $1$ for the denominator: $5\to4$
2. Reduce the count of the item taken by $1$: red $3\to2$
3. Write the fractions: red $\frac{2}{4}$, blue $\frac{2}{4}$

True or false? When the same biased coin is flipped twice, the probabilities on the second set of branches are the same as on the first set.

True.
The flips are independent, so the coin's probabilities are unchanged.

Name two things to check when asked to write down what is wrong with a given probability tree diagram.

Each set of branches from one point must sum to $1$, and each probability must be on the branch with the matching label (not reversed).

How do you find the probability of one outcome followed by another from a tree diagram?

e.g. P(train to work late) $=0.13$, P(train home late) $=0.06$; find P(both late)

Multiply the probabilities along the branches.
$0.13\times0.06=0.0078$

What are the steps to find the probability of 'exactly one' from a tree diagram?

e.g. a spinner lands on 2 with probability $\frac{1}{3}$ and is spun twice

1. Multiply along one order: $\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$
2. Multiply along the other order: $\frac{2}{3}\times\frac{1}{3}=\frac{2}{9}$
3. Add: $\frac{2}{9}+\frac{2}{9}=\frac{4}{9}$

What are the steps to estimate the probability of one result then another from the results of an experiment?

e.g. a drawing pin dropped 150 times landed point up 50 times and point down 100 times; find P(up then down)

1. Estimate each probability using the overall total: $\frac{50}{150}$ and $\frac{100}{150}$
2. Multiply them: $\frac{50}{150}\times\frac{100}{150}=\frac{2}{9}$

True or false? A fair dice is thrown twice, so the probability of getting a 6 on both throws is $\frac{1}{6}+\frac{1}{6}=\frac{2}{6}$.

False.
You multiply, not add: $\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}$.

What assumption are you making when you multiply two probabilities together to find the probability of both events happening?

That the events are independent — the outcome of one does not affect the probability of the other.

What is the quickest way to find the probability of 'at least one' from a tree diagram?

e.g. Dan plays two games and wins each with probability $0.3$; find P(at least one win)

$1 -$ P(neither), i.e. $1 - (0.7 \times 0.7) = 0.51$

True or false? To find the probability of 'the same on both' you multiply along each matching path and then add those results.

True. e.g. P(both heads) $+$ P(both tails).