IB Maths AA SL 4.6 Combined Conditional and Independent Events Questions

Practise IB Mathematics SL 4.6 by applying combined, conditional and independent events methods to questions from the Question Bank.

Exam points

Represent combined events with Venn, tree and sample-space diagrams or outcome tables, and calculate P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Use conditional probability and independence for dependent or independent events, handling with- and without-replacement assumptions explicitly.

[Maximum number: 6]

A nationwide study on reaction time is conducted on participants in two age groups. The participants in Group X are less than 40 years old. Their reaction times are normally distributed with mean 0.489 seconds and standard deviation 0.07 seconds.

A randomly selected participant has a reaction time greater than 0.65 seconds. Find the probability that the participant is in Group X.

correct work for P (group X and t>0.65 ) or P (group Y and t>0.65 ) (may be seen anywhere)
eg P( group X $) \times \mathrm{P}(t>0.65 \mid \mathrm{X}), \mathrm{P}(X \cap t>0.65)=0.0107 \times 0.38(=0.004075)$ , $\mathrm{P}(Y \cap t>0.65)=0.396 \times 0.62$
recognizing conditional probability (seen anywhere)
eg $\mathrm{P}(\mathrm{X} \mid t>0.65), \mathrm{P}(A \mid B)=\frac{\mathrm{P}(A \cap B)}{\mathrm{P}(B)}$
valid approach to find P(t>0.65)

Figure for Answer to Question IB Maths AA SL 4.6 Combined Conditional and Independent Events Questions question 1 — IB Maths AA SL

correct work for P(t>0.65)
eg $\quad 0.0107 \times 0.38+0.396 \times 0.62,0.249595$
correct substitution into conditional probability formula
eg $\frac{0.0107 \times 0.38}{0.0107 \times 0.38+0.396 \times 0.62}, \frac{0.004075}{0.249595}$
0.016327
$\mathrm{P}(\mathrm{X} \mid t>0.65)=0.0163270$

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IB Maths AA SL 4.6 Combined Conditional and Independent Events Questions question 1