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# Probability worded problem

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Hey there! I’m having trouble with questing 25c, any help would be appreciated.

Thankyou!

Guest Jul 23, 2018

#1
+20681
+1

Probability worded problem

Jane has three bags of lollies. In bag 1 there are three mints and three toffees,
in bag 2 there are three mints and two toffees and in bag 3 there are two mints and one toffee.
Jane selects a bag at random and then selects a lolly at random.

c the probability hat jane chose bag 1, given that she selects a mint.

$$\text{P(bag 1) = \dfrac13 } \\ \text{P(mint/bag 1) = \dfrac36} \\ \text{P(mint form bag 1) = P(bag 1)P(mint/bag 1) = \left(\dfrac13\right)\left(\dfrac36\right) = \dfrac16 }$$

heureka  Jul 23, 2018
#1
+20681
+1

Probability worded problem

Jane has three bags of lollies. In bag 1 there are three mints and three toffees,
in bag 2 there are three mints and two toffees and in bag 3 there are two mints and one toffee.
Jane selects a bag at random and then selects a lolly at random.

c the probability hat jane chose bag 1, given that she selects a mint.

$$\text{P(bag 1) = \dfrac13 } \\ \text{P(mint/bag 1) = \dfrac36} \\ \text{P(mint form bag 1) = P(bag 1)P(mint/bag 1) = \left(\dfrac13\right)\left(\dfrac36\right) = \dfrac16 }$$

heureka  Jul 23, 2018
#2
+1225
+1

Solution:

$$\text {For these types of question, use the Total Probability Theorem/Formula.}\\ \mathbb{P}(B_1) =\mathbb{P}(B_2) =\mathbb{P}(B_3) = \dfrac{1}{3}\\ \text {Probability of mint from Bag} \tiny \text {#} \\ \mathbb{P}(M|B_1) = \dfrac{1}{2}\\ \mathbb{P}(M|B_2) = \dfrac{3}{5}\\ \mathbb{P}(M|B_3) = \dfrac{2}{3}\\ \;\\ \mathbb{P}(B_1|M) = \Big (\mathbb{P}(B_1) \cdot \mathbb{P}(M|B_1)\Big)\div \Big(\mathbb{P}(B_1) \mathbb{P}(M|B1) + \mathbb{P}(B_2)\mathbb{P}(M|B_2) +\mathbb{P} P(B_3) \mathbb{P}(M|B_3)\Big)$$

$$\text {The probability that Jane chose bag 1, given that she selects a mint is equal to the probability}\\ \text {Jane chose a mint from bag #1 divided by the probability that she selected a mint.}\\$$

$$\mathbb{P}(B_1|M) = \dfrac{\Big(\dfrac{1}{3} \cdot \dfrac{1}{2}\Big) }{\Big(\dfrac{1}{3} \cdot \dfrac{1}{2}\Big) + \Big(\dfrac{1}{3} \cdot \dfrac{3}{5}\Big) +\Big( \dfrac{1}{3} \cdot \dfrac{2}{3}\Big)}\\ \;\\ \hspace{1.8cm}=\dfrac {15}{53}\\ \;\\ \hspace{1.8cm} \approx 0.28301\\$$

GA

GingerAle  Jul 24, 2018
edited by GingerAle  Jul 24, 2018