Skip to main content

Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 3.2.4 Mutually Exclusive and Independent Events

Subsubsection 3.2.4.1 Mutually Exclusive

Activity 3.2.4.

\({\color{black} \textbf{Work in groups}}\)
🔗
  1. Define Mutually exclusive events
    🔗
    🔗
  2. State one example of Two events that are mutually exclusive
    🔗
    🔗
  3. In a class of 40 students, 18 take French, 22 take German, and no student takes both.
    1. What is the probability that a randomly selected student takes French or German?
      🔗
      🔗
    2. Are the events “taking French” and “taking German” mutually exclusive? Explain.
      🔗
      🔗
    🔗
    🔗
  4. Compare and discuss answers with other groups
    🔗
    🔗
🔗
🔗
\({\color{black} \textbf{Key Takeaway}}\)
🔗
Two events are mutually exclusive if they cannot occur at the same time.
🔗
This means that if one event happens, the other cannot.
🔗
If \(\textbf{A} \) and \(\text{B}\) are mutually exclusive events, then;
\begin{gather*} \textbf{P(A and B) = 0} \end{gather*}
🔗
\begin{gather*} \textbf{P}(\textbf{A} \cap \textbf{B} ) \, = \, 0 \end{gather*}
🔗
The probability of either \(\textbf{A}\) or \(\textbf{B}\) occurring is;
\begin{gather*} \textbf{P(A or B) = P(A) + P(B)} \end{gather*}
🔗
\(\text{Example} \)
  • A traffic light being red and the same traffic light being green at the exact same time.
    🔗
    🔗
  • You are sleeping and you are wide awake at the exact same moment.
    🔗
    🔗
  • A door being open and the same door being closed at the same time.
    🔗
    🔗
🔗

Example 3.2.11.

Roll a fair six-sided die, what is the probability of rolling either a \(3\) or a \(5\text{?}\)
🔗
Solution.
Sample Space
\begin{gather*} \textbf{S = {1, 2, 3, 4, 5, 6}} \end{gather*}
🔗
Favorable Outcomes
  • \(\displaystyle \textbf{P(3)} = \frac{1}{6}\)
    🔗
    🔗
  • \(\displaystyle \textbf{P(5)} = \frac{1}{6}\)
    🔗
    🔗
🔗
Since rolling a 3 and rolling a 5 are mutually exclusive events;
\begin{gather*} \textbf{P(3 or 5)} = \textbf{P(3) + P(5)} \end{gather*}
\begin{gather*} = \frac{1}{6} + \frac{1}{6} \end{gather*}
\begin{gather*} = \frac{2}{6} = \frac{1}{3} \end{gather*}
🔗
the probability of rolling a \(3\) or \(5\) is \(\frac{1}{3}\) or \(33.33\%\)
🔗
🔗
🔗

Example 3.2.12.

A card is drawn from a standard deck of 52 playing cards. Let Event A be drawing a Heart and Event B be drawing a Spade. Are these events mutually exclusive? Explain
🔗
  1. Find \(\textbf{P(A) and P(B)} \)
    🔗
    🔗
  2. Calculate \(\textbf{P(A or B)}\)
    🔗
    🔗
  3. What is \(\textbf{P}(\textbf{A} \, \cap \, \textbf{B})\)
    🔗
    🔗
🔗
Solution.
Two events are mutually exclusive if they cannot happen at the same time. Since a card cannot be both a Heart and a Spade, the events are mutually exclusive.
🔗
  1. We calculate the probability of drawing a Heart or a Spade.
    🔗
    Since there are \(\textbf{13 Hearts}\) in the deck,
    🔗
    \begin{gather*} \textbf{P(A)} \, = \, \frac{13}{52} \end{gather*}
    🔗
    Since there are \(\textbf{13 Spades}\) in the deck,
    🔗
    \begin{gather*} \textbf{P(B)} \, = \, \frac{13}{52} \end{gather*}
    🔗
    🔗
  2. For mutually exclusive events, we use,
    🔗
    \begin{gather*} \textbf{P(A or B) = P(A) + P(B)} \end{gather*}
    🔗
    \begin{gather*} = \frac{13}{52} + \frac{13}{52} \end{gather*}
    🔗
    \begin{gather*} = \frac{26}{52} \end{gather*}
    🔗
    \begin{gather*} = \frac{1}{2} \end{gather*}
    🔗
    🔗
  3. Since these events are mutually exclusive, their intersection is zero:
    🔗
    \begin{gather*} \textbf{P} (\textbf{A} \cap \textbf{B}) = 0 \end{gather*}
    🔗
    🔗
🔗
🔗
🔗

Exercises Exercises

1.
A student at Khungu Senior Secondary School tosses a coin once. Are the events "getting heads" and "getting tails" mutually exclusive?
🔗
🔗
2.
A person selects one piece of fruit from a bowl containing apples, bananas, and oranges.Is selecting an apple and selecting a banana mutually exclusive events?
🔗
🔗
3.
A student recorded their method of transport to school for 10 days. The methods used were:
\begin{gather*} \textbf{Walk, Bus, Bus, Walk, Bike, Walk, Bus, Bike, Walk, Bus} \end{gather*}
Let Event A = "The student walked to school" and Event B = "The student took the bus to school"
  1. Are events A and B mutually exclusive? Explain your answer.
    🔗
    🔗
  2. What is the probability that the student either walked or took the bus to school on a randomly chosen day?
    🔗
    🔗
🔗
🔗
4.
A card is drawn from a standard 52-card deck.
  1. Are the events "drawing a diamond" and "drawing a club" mutually exclusive?
    🔗
    🔗
  2. What is the probability of drawing either a diamond or a club?
    🔗
    🔗
🔗
🔗
5.
In a lottery competition, there are five cards labelled A, B, C, D, and F. A player must pick only one card to enter the competition.
  1. Are the events "picking card A" and "picking card F" mutually exclusive?
    🔗
    🔗
  2. What is the probability of either picking card A or picking card F?
    🔗
    🔗
🔗
🔗
🔗
🔗

Subsubsection 3.2.4.2 Independent Events

Activity 3.2.5.

\({\color{black} \textbf{Work in groups}}\)
🔗
  1. Define Independent events
    🔗
    🔗
  2. State one example of two events that are independent
    🔗
    🔗
  3. A student can choose to join either the Science Club or the Drama Club, but not both.
    1. If the probability of joining Science Club is \(40\%\) and Drama Club is \(30\%\) , what is the probability that a student joins either club?
      🔗
      🔗
    2. Are these events mutually exclusive or independent? Explain.
      🔗
      🔗
    🔗
    🔗
  4. Compare and discuss answers with other groups
    🔗
    🔗
🔗
🔗
\({\color{black} \textbf{Key Takeaway}}\)
🔗
Two events are independent if the occurrence of one does not affect the probability of the other occurring.
🔗
If \(A\) and \(B\) are independent events, then;
\begin{gather*} P(\textbf{A and B}) = \textbf{P(A)} \times \textbf{P(B)} \end{gather*}
🔗
For events \(\textbf{A}\) and \(\textbf{B}\)
🔗
\begin{gather*} \textbf{P}(\textbf{A} \cap \textbf{B}) \, = \, \textbf{P(A)} \times \textbf{P(B)} \end{gather*}
🔗
This means the probability of both events occurring together is the product of their individual probabilities.
🔗
\(\text{Example}\)
  • A student bringing a lunch from home and another student buying a lunch from the cafeteria.
    🔗
    🔗
  • A student answering a question correctly in english class and another student dropping their pencil in science class.
    🔗
    🔗
🔗

Example 3.2.13.

A coin is flipped, and a six-sided die is rolled. What is the probability of getting heads and rolling a 6?
🔗
Solution.
  1. the Sample Space is
    🔗
    Possible coin outcomes \(\textbf{{H, T}}\)
    🔗
    Possible die outcomes \(\textbf{{1, 2, 3, 4, 5, 6}}\)
    🔗
    🔗
  2. Favorable Outcomes
    🔗
    \(\textbf{P(H)} = \frac{1}{2}\)
    🔗
    \(\textbf{P(6)} = \frac{1}{6}\)
    🔗
    🔗
  3. Since flipping the coin and rolling the die are independent events
    🔗
    \begin{gather*} P(\textbf{H and 6}) = \textbf{P(H)} \times \textbf{P(6)} \end{gather*}
    🔗
    \begin{gather*} = \frac{1}{2} \times \frac{1}{6} \end{gather*}
    🔗
    \begin{gather*} = \frac{1}{12} \end{gather*}
    🔗
    🔗
🔗
the probability of getting heads and a 6 is \(\frac{1}{12}\) or \(8.33\%\)
🔗
🔗
🔗

Example 3.2.14.

The probability that it rains on a given day is \(40\%\text{,}\) and the probability that a person is late to work is \(20\%\text{.}\)
🔗
Let \(\textbf{P(R)}\) represent the probability that it rains and \(\textbf{P(L)}\) be the probability that the person is late. The compliment of an event is the probability that it does not happen. Therefore, \(\textbf{P}(\textbf{R}^{c})\) will represent the probability that it does not rain and \(\textbf{P}(\textbf{L}^{c})\) the probability that the person is not late.
🔗
  1. Find the probability that it rains \(\textbf{P(R)}\) .
    🔗
    🔗
  2. Find the probability that the person is late \(\textbf{P(L)}\text{.}\)
    🔗
    🔗
  3. Find the probability that it rains and the person is late.
    🔗
    🔗
  4. Find the probability that it does not rain but the person is late.
    🔗
    🔗
  5. Find the probability that it rains or the person is late.
    🔗
    🔗
🔗
Solution.
  1. The probability that it rains is
    🔗
    \begin{gather*} \textbf{P(R) = 0.4} \end{gather*}
    🔗
    The probability that it rains is \(\textbf{0.4 (or 40%)}\text{.}\)
    🔗
    🔗
  2. The probability that the person is late is
    🔗
    \begin{align*} \textbf{P(L) = 0.2} \amp \end{align*}
    🔗
    The probability that the person is late is \(0.2 \) or \(( 20\%)\text{.}\)
    🔗
    🔗
  3. Since rain and being late are independent,
    🔗
    \begin{gather*} \textbf{P}(\textbf{R} \, \cap \textbf{L}) \, = \, \textbf{P(R)} \times \textbf{P(L)} \end{gather*}
    🔗
    \begin{gather*} \textbf{P}(\textbf{R} \, \cap \textbf{L}) \, = \, 0.4 \times 0.2 \, = \, 0.08 \end{gather*}
    🔗
    The probability that it rains and the person is late is \(\textbf{0.08 (or 8%)}\text{.}\)
    🔗
    🔗
  4. The probability that it does not rain is
    🔗
    \begin{gather*} \textbf{P}(\textbf{R}^{c}) \, = \, \textbf{1 - P(R)} = 1 \, - \, 0.4 \, = \, 0.6 \end{gather*}
    🔗
    Since rain and being late are independent
    🔗
    \begin{gather*} \textbf{P}(\textbf{R}^{c} \cap L) = \textbf{P}(\textbf{R}^{c}) \times \textbf{P(L)} \end{gather*}
    🔗
    \begin{gather*} \textbf{P}(\textbf{R}^{c} \cap L) = 0.6 \, \times \, 0.2 \, = \, 012 \end{gather*}
    🔗
    The probability that it does not rain but the person is late is \(0.12 \) or \(( 12\%)\text{.}\)
    🔗
    🔗
  5. The probability that it rains or the person is late is
    🔗
    \begin{gather*} \textbf{P}(\textbf{R} \cup \textbf{L}) = \textbf{P(R)} \, + \, \textbf{P(R)} \, - \, \textbf{P}(\textbf{R} \cup \textbf{L}) \end{gather*}
    🔗
    \begin{gather*} \textbf{P}(\textbf{R} \cup \textbf{L}) = 0.4 \, + \, 0.2 \, - \, 0.08 \end{gather*}
    🔗
    \begin{gather*} = \, 0.52 \end{gather*}
    🔗
    The probability that it rains or the person is late is \(0.52 \) or \(( 52\%)\text{.}\)
    🔗
    🔗
🔗
🔗
🔗

Exercises Exercises

1.
A coin is tossed twice. What is the probability of getting heads on both tosses?
🔗
🔗
2.
A die is rolled, and a coin is tossed. What is the probability of rolling a 6 on the die and getting tails on the coin?
🔗
🔗
3.
A bakery in Makongeni produces cakes. The probability that a cake is decorated with chocolate icing is \(0.7\text{.}\) If two cakes are made independently, what is the probability that both cakes are decorated with chocolate icing?
🔗
🔗
4.
A seed has a \(60%\) chance of germinating. If two seeds are planted independently, what is the probability that both seeds germinate?
🔗
🔗
🔗
🔗

Checkpoint 3.2.15.

🔗

Checkpoint 3.2.16.

🔗

Checkpoint 3.2.17.

🔗
🔗
PrevTopNext