Sec 4 E-Math · Probability

Independent events questions with answers

Practice questions on independent events, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

A fair coin is tossed and a fair 8-sided die, numbered $1$ to $8$ , is thrown. Find the probability of getting a head and a $5$ .

Step 1 of 1

Find each probability, then combine them: both must happen

P(\text{head}) = \dfrac{1}{2}

P(5) = \dfrac{1}{8}

\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{8}

P = \dfrac{1}{16}

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Questions

Question 1Coin and die
A fair coin is tossed and a fair 4-sided die, numbered $1$ to $4$ , is thrown. Find the probability of getting a head and a $3$ .
Show the worked solution
1. Step 1: Find each probability, then combine them: both must happen $P(\text{head}) = \dfrac{1}{2}$ $P(3) = \dfrac{1}{4}$ $\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{4}$ $P = \dfrac{1}{8}$
Question 2Coin and die
A fair coin is tossed and a fair 8-sided die, numbered $1$ to $8$ , is thrown. Find the probability of getting a tail and a $5$ .
Show the worked solution
1. Step 1: Find each probability, then combine them: both must happen $P(\text{tail}) = \dfrac{1}{2}$ $P(5) = \dfrac{1}{8}$ $\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{8}$ $P = \dfrac{1}{16}$
Question 3Both, one but not the other, neither
The probability that it rains tomorrow is $0.2$ and the probability that the bus is late is $0.9$ . The two events are independent. Find the probability that it rains tomorrow but the bus is not late.
Show the worked solution
1. Step 1: P(not happening) = 1 − P(happening) $P(\text{second does not occur}) = 1 - 0.9$ $\phantom{P(\text{second does not occur})} = 0.1$
2. Step 2: Combine the two probabilities: both must happen $\mathstrut P = 0.2 \times 0.1$ $P = 0.02$
Question 4Both, one but not the other, neither
The probability that Ali passes his driving test is $0.9$ and the probability that Ben passes his driving test is $0.15$ . The two events are independent. Find the probability that neither event occurs.
Show the worked solution
1. Step 1: P(not happening) = 1 − P(happening) $P(\text{first does not occur}) = 1 - 0.9$ $\phantom{P(\text{first does not occur})} = 0.1$ $P(\text{second does not occur}) = 1 - 0.15$ $\phantom{P(\text{second does not occur})} = 0.85$
2. Step 2: Combine the two probabilities: both must happen $\mathstrut P = 0.1 \times 0.85$ $P = 0.085$
Question 5At least one
The probability that Ali passes his driving test is $0.4$ and the probability that Ben passes his driving test is $0.5$ . The two events are independent. Find the probability that at least one of the two events occurs.
Show the worked solution
1. Step 1: The probability that each event does not occur $P(\text{first does not occur}) = 1 - 0.4$ $\phantom{P(\text{first does not occur})} = 0.6$ $P(\text{second does not occur}) = 1 - 0.5$ $\phantom{P(\text{second does not occur})} = 0.5$
2. Step 2: Neither occurs: combine the two $\mathstrut P(\text{neither}) = 0.6 \times 0.5$ $P(\text{neither}) = 0.3$
3. Step 3: At least one occurs: use the result for neither $\mathstrut P = 1 - 0.3$ $P = 0.7$
Question 6At least one
The probability that Ali passes his driving test is $0.25$ and the probability that Ben passes his driving test is $0.4$ . The two events are independent. Find the probability that at least one of the two events occurs.
Show the worked solution
1. Step 1: The probability that each event does not occur $P(\text{first does not occur}) = 1 - 0.25$ $\phantom{P(\text{first does not occur})} = 0.75$ $P(\text{second does not occur}) = 1 - 0.4$ $\phantom{P(\text{second does not occur})} = 0.6$
2. Step 2: Neither occurs: combine the two $\mathstrut P(\text{neither}) = 0.75 \times 0.6$ $P(\text{neither}) = 0.45$
3. Step 3: At least one occurs: use the result for neither $\mathstrut P = 1 - 0.45$ $P = 0.55$

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Worked example

A fair coin is tossed and a fair 8-sided die, numbered

1

8

, is thrown. Find the probability of getting a head and a

5

Step 1 of 1

Find each probability, then combine them: both must happen

P(\text{head}) = \dfrac{1}{2}

P(5) = \dfrac{1}{8}

\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{8}

P = \dfrac{1}{16}

Questions

Question 1Coin and die

A fair coin is tossed and a fair 4-sided die, numbered $1$ to $4$ , is thrown. Find the probability of getting a head and a $3$ .

Show the worked solution

Step 1: Find each probability, then combine them: both must happen $P(\text{head}) = \dfrac{1}{2}$ $P(3) = \dfrac{1}{4}$ $\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{4}$ $P = \dfrac{1}{8}$

Question 2Coin and die

A fair coin is tossed and a fair 8-sided die, numbered $1$ to $8$ , is thrown. Find the probability of getting a tail and a $5$ .

Show the worked solution

Step 1: Find each probability, then combine them: both must happen $P(\text{tail}) = \dfrac{1}{2}$ $P(5) = \dfrac{1}{8}$ $\mathstrut P = \dfrac{1}{2} \times \dfrac{1}{8}$ $P = \dfrac{1}{16}$

Question 3Both, one but not the other, neither

The probability that it rains tomorrow is $0.2$ and the probability that the bus is late is $0.9$ . The two events are independent. Find the probability that it rains tomorrow but the bus is not late.

Show the worked solution

Step 1: P(not happening) = 1 − P(happening) $P(\text{second does not occur}) = 1 - 0.9$ $\phantom{P(\text{second does not occur})} = 0.1$
Step 2: Combine the two probabilities: both must happen $\mathstrut P = 0.2 \times 0.1$ $P = 0.02$

Question 4Both, one but not the other, neither

The probability that Ali passes his driving test is $0.9$ and the probability that Ben passes his driving test is $0.15$ . The two events are independent. Find the probability that neither event occurs.

Show the worked solution

Step 1: P(not happening) = 1 − P(happening) $P(\text{first does not occur}) = 1 - 0.9$ $\phantom{P(\text{first does not occur})} = 0.1$ $P(\text{second does not occur}) = 1 - 0.15$ $\phantom{P(\text{second does not occur})} = 0.85$
Step 2: Combine the two probabilities: both must happen $\mathstrut P = 0.1 \times 0.85$ $P = 0.085$

Question 5At least one

The probability that Ali passes his driving test is $0.4$ and the probability that Ben passes his driving test is $0.5$ . The two events are independent. Find the probability that at least one of the two events occurs.

Show the worked solution

Step 1: The probability that each event does not occur $P(\text{first does not occur}) = 1 - 0.4$ $\phantom{P(\text{first does not occur})} = 0.6$ $P(\text{second does not occur}) = 1 - 0.5$ $\phantom{P(\text{second does not occur})} = 0.5$
Step 2: Neither occurs: combine the two $\mathstrut P(\text{neither}) = 0.6 \times 0.5$ $P(\text{neither}) = 0.3$
Step 3: At least one occurs: use the result for neither $\mathstrut P = 1 - 0.3$ $P = 0.7$

Question 6At least one

The probability that Ali passes his driving test is $0.25$ and the probability that Ben passes his driving test is $0.4$ . The two events are independent. Find the probability that at least one of the two events occurs.

Show the worked solution

Step 1: The probability that each event does not occur $P(\text{first does not occur}) = 1 - 0.25$ $\phantom{P(\text{first does not occur})} = 0.75$ $P(\text{second does not occur}) = 1 - 0.4$ $\phantom{P(\text{second does not occur})} = 0.6$
Step 2: Neither occurs: combine the two $\mathstrut P(\text{neither}) = 0.75 \times 0.6$ $P(\text{neither}) = 0.45$
Step 3: At least one occurs: use the result for neither $\mathstrut P = 1 - 0.45$ $P = 0.55$