IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Static and Probability-Combined events
Topic :Static and Probability- Weightage : 21 %
All Questions for Topic : Sampling techniques,Data manipulation and misinterpretation,Graphical representations (including: bivariate graphs, scatter graphs, box plots, cumulative frequency graphs)
,Lines of best fit,Data processing: quartiles and percentiles,Measures of dispersion: interquartile range,Correlation, qualitative handling,Relative frequency,Response rates,Sets, including notation and operations up to three sets,Probability with Venn diagrams, tree diagrams and sample spaces,Mutually exclusive events,Combined events
Question (4 Marks)
What is the probability that you and your friend will each draw a pen of the same color from the box, considering that you pick first, replace the pen, and then your friend picks from the same box? The box contains 3 red pens, 4 black pens, and 2 green pens. Determine the likelihood of this outcome.
▶️Answer/Explanation
Ans:
Let \(N\) be the total number of pens in the box, \(n_r\) be the number of red pens, \(n_b\) be the number of black pens, and \(n_g\) be the number of green pens.
So, \(N = n_r + n_b + n_g = 3 + 4 + 2 = 9\).
The probability of you selecting a red pen is given by:
\[P(\text{You select red}) = \frac{n_r}{N} = \frac{3}{9} = \frac{1}{3}.\]
The probability of your friend selecting a red pen is also:
\[P(\text{Friend selects red}) = \frac{n_r}{N} = \frac{3}{9} = \frac{1}{3}.\]
Since you and your friend are selecting independently, the probability of both events happening (you selecting a red pen and your friend selecting a red pen) is the product of their individual probabilities:
\[P(\text{Both select red}) = P(\text{You select red}) \times P(\text{Friend selects red}) = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}.\]
Similarly, the probability of both selecting a black pen is:
\[P(\text{Both select black}) = P(\text{You select black}) \times P(\text{Friend selects black}) = \frac{4}{9} \times \frac{4}{9} = \frac{16}{81}.\]
And the probability of both selecting a green pen is:
\[P(\text{Both select green}) = P(\text{You select green}) \times P(\text{Friend selects green}) = \frac{2}{9} \times \frac{2}{9} = \frac{4}{81}.\]
To get the overall probability of both you and your friend selecting the same color pen, you add the probabilities of all three scenarios:
\[P(\text{Both select same color pen}) = P(\text{Both select red}) + P(\text{Both select black}) + P(\text{Both select green})\]
\[= \frac{1}{9} + \frac{16}{81} + \frac{4}{81} = \frac{29}{81}.\]
So, the probability that you and your friend will each draw a pen of the same color from the box is \(\frac{29}{81}\).