Compound events
Picking a Leap Card top-up and then a bus or Luas, drawing two cards, or first choosing a machine and then checking a part: these are compound experiments, made up of several stages. Before distinguishing whether the stages are independent or dependent (the next two sections), learn to recognise the structure common to all of them. Back at the start: a coin flip followed by another is a two-stage experiment, and you will work out its probability below: each stage contributes a probability, and you have to combine them correctly.
Multi-stage experiments
We flip two coins independently. Calculate the probability of obtaining exactly two tails.
We flip two coins independently. Calculate the probability of obtaining exactly two tails.
We roll a fair six-sided die (with faces numbered from $1$ to $6$) and flip a coin. What is the probability of obtaining an even number on the die and a head on the coin?
We roll a fair six-sided die (with faces numbered from $1$ to $6$) and flip a coin. What is the probability of obtaining an even number on the die and a head on the coin?
We spin the two independent spinners shown in the figure. Calculate the probability that the first spinner lands on the number $3$ and the second spinner lands on green…
We spin the two independent spinners shown in the figure. Calculate the probability that the first spinner lands on the number $3$ and the second spinner lands on green (V).
The general product rule
In a restaurant, $60\%$ of customers order the set menu and $40\%$ order à la carte. Of those ordering the set menu, $70\%$ have dessert. Of those ordering à la carte,…
In a restaurant, $60\%$ of customers order the set menu and $40\%$ order à la carte. Of those ordering the set menu, $70\%$ have dessert. Of those ordering à la carte, $50\%$ have dessert. Calculate the probability that a customer chosen at random orders the set menu and has dessert.
In a Year 11 class there are $12$ boys and $18$ girls. It is known that $4$ boys and $6$ girls wear glasses. If we choose a student at random, what is the probability…
In a Year 11 class there are $12$ boys and $18$ girls. It is known that $4$ boys and $6$ girls wear glasses. If we choose a student at random, what is the probability that they are a girl who wears glasses?
Three cards are drawn consecutively and without replacement from a deck of $40$ cards. Calculate the probability of obtaining three cards of the suit of coins.
Three cards are drawn consecutively and without replacement from a deck of $40$ cards. Calculate the probability of obtaining three cards of the suit of coins.
Two paths to the same outcome
When the experiment has a first stage with several options and each option leads to a different second stage, you have to add the probabilities of all the paths that lead to the desired outcome.
We have two urns. Urn $A$ contains $4$ white balls and $2$ black balls. Urn $B$ contains $3$ white balls and $5$ black balls. We flip a fair coin; if it lands on heads,…
We have two urns. Urn $A$ contains $4$ white balls and $2$ black balls. Urn $B$ contains $3$ white balls and $5$ black balls. We flip a fair coin; if it lands on heads, we draw a ball from urn $A$, and if it lands on tails, we draw a ball from urn $B$. Find the probability that the drawn ball is white.
An electronic component factory has two machines. Machine $M_1$ produces $60\%$ of the components and machine $M_2$ the remaining $40\%$. It is known that $2\%$ of the…
An electronic component factory has two machines. Machine $M_1$ produces $60\%$ of the components and machine $M_2$ the remaining $40\%$. It is known that $2\%$ of the components produced by $M_1$ are defective, while $5\%$ of those produced by $M_2$ are defective. If a component is selected at random from the warehouse, what is the probability that it is defective?
"At least one" in compound experiments
Two basketball players, Aoife and Niamh, each take a free throw. The probability that Aoife scores is $0.8$ and the probability that Niamh scores is $0.7$. Their shots…
Two basketball players, Aoife and Niamh, each take a free throw. The probability that Aoife scores is $0.8$ and the probability that Niamh scores is $0.7$. Their shots are independent events. Calculate the probability that at least one of them scores her free throw.
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. Water flows from left to right. For water to reach the end, it must pass through valve $A$…
The following diagram shows a pipe system with three valves $A$, $B$ and $C$. Water flows from left to right. For water to reach the end, it must pass through valve $A$ or through the lower branch containing valves $B$ and $C$ in series. The probabilities of the valves being open are $P(A) = 0.7$, $P(B) = 0.8$ and $P(C) = 0.9$. Knowing that the operation of each valve is independent, calculate the probability that water successfully passes through the system.