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(P) 6 The events $A$ and $B$ are such that $\mathrm{P}(A)=\frac{1}{3}$ and $\mathrm{P}(B)=\frac{1}{4}, \mathrm{P}(A$ or $B$ or both $)=\frac{1}{2}$.
a Represent these probabilities on a Venn diagram.
b Show that $A$ and $B$ are independent.
(E) 7 The Venn diagram shows the number of students who like either cricket ( $C$ ), football $(F)$ or swimming $(S)$.
a Which two sports are mutually exclusive?
(1 mark)
b Determine whether the events 'likes cricket' and 'likes football' are independent.
(3 marks)
(E/P) 8 For events $J$ and $K, \mathrm{P}(J$ or $K$ or both $)=0.5, \mathrm{P}(K$ but not $J)=0.2$ and $\mathrm{P}(J$ but not $K)=0.25$.
a Draw a Venn diagram to represent events $J$ and $K$ and the sample space $S$.
(3 marks)
b Determine whether events $J$ and $K$ are independent.
(3 marks)
(E) 9 A survey of a group of students revealed that $85 \%$ have a mobile phone, $60 \%$ have an MP3 player and $5 \%$ have neither phone nor MP3 player.
a Find the proportion of students who have both gadgets.
(2 marks)
b Draw a Venn diagram to represent this information.
(3 marks)
c A student is chosen at random. Find the probability that they only own a mobile phone.
(2 marks)
d Are the events 'own a mobile phone' and 'own an MP3 player' independent? Justify your answer.
(3 marks)
(E/P) 10 The Venn diagram shows the probabilities that a group of children like cake ( $A$ ) or crisps (B).
Determine whether the events 'like cake' and 'like crisps' are independent.
(3 marks)
(E/P) 11 A computer game has three levels and one of the objectives of every level is to collect a diamond. The probability that Becca collects a diamond on the first level is $\frac{4}{5}$, the second level is $\frac{2}{3}$ and the third level is $\frac{1}{2}$. The events are independent.
a Draw a tree diagram to represent Becca collecting diamonds on the three levels of the game.
(4 marks)
b Find the probability that Becca:
i collects all three diamonds
(2 marks)
ii collects only one diamond.
(3 marks)
c Find the probability that she collects at least two diamonds each time she plays.
(3 marks)
(P) 12 In a factory, machines $A, B$ and $C$ produce electronic components. Machine $A$ produces $16 \%$ of the components, machine $B$ produces $50 \%$ of the components and machine $C$ produces the rest. Some of the components are defective. Machine $A$ produces $4 \%$, machine B $3 \%$ and machine C $7 \%$ defective components.
a Draw a tree diagram to represent this information.
b Find the probability that a randomly selected component is:
i produced by machine $B$ and is defective
ii defective.

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The probability of selecting a red bead is given by $ P(R) = \frac{7}{15} $ and for a blue or green bead is $ P(B \cup G) = \frac{3 + 5}{15} = \frac{8}{15} $. ‖ ‖ step 2 ⋮ The probability of the first two beads being red and the third being blue or green is calculated as: $ P(R, R, B \cup G) = P(R) \times P(R) \times P(B \cup G) = \left(\frac{7}{15}\right) \times \left(\frac{7}{15}\right) \times \left(\frac{8}{15}\right) = \frac{392}{3375} $. ‖ ‖ step 3 ⋮ For part b, the probability of choosing one red, one blue, and one green bead can be calculated using combinations. The total arrangements are $ 3! = 6 $ and the probability is $ P(R) \times P(B) \times P(G) = \frac{7}{15} \times \frac{3}{15} \times \frac{5}{15} $. Thus, the total probability is $ 6 \times \left(\frac{7}{15} \times \frac{3}{15} \times \frac{5}{15}\right) = \frac{630}{3375} = \frac{42}{225} $. ‖ ∻Answer∻ ⚹ Part a: $ \frac{392}{3375} $, Part b: $ \frac{42}{225} $ ⚹ ∻Key Concept∻ ⚹Probability of independent events and combinations in probability.⚹ ∻Explanation∻ ⚹The calculations show how to find the probability of specific outcomes when selecting beads, using the principles of probability and combinations.⚹◊What is the formula to determine if two events are independent in probability?⍭ Generate me a similar question◊

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