probability problems

[匿名]

2013-02-09 11:43 am

1. Anna enjoys making artworks out of clay. 65% of her artworks are about fruit, of which 30% are large-sized, 65% are medium-sized, and the rest are small-sized. Her remaining artworks are about furniture, of which 25% are large-sized, 60% are medium-sized and 15% are small-sized. suppose an artwork is randomly selected.

if Anna changes the ratio of her artworks about fruit and furniture such that 10% of the artworks are small-sized, what proportion of all her artworks should be about fruit?
answer 0.5

2. three famous hair stylists work for a hair salon. the probabilities that a customer goes for stylists A, B and C are 0.35, 0.25 and 0.4 respectively. the probabilities that stylists A, B and C are available for providing service are 0.8, 0.65 and 0.3 respectively. All the customers' choices of the stylists and their availability are independent.

If 3 customers are selected at random, what is the probability that only one of them is served by his/her favourite stylist?
answer 0.32

please list out the steps needed to get the answers. thanks

回答 (1)

fooks

2013-02-09 6:17 pm

✔ 最佳答案

1.
Let n (n ≤ 1) be the new proportion of her artworks about fruit.

Proportion of small-sized artworks about fruit
= n x (1 - 30% - 65%)
= 0.05n

Proportion of small-sized artworks about furniture
= (1 - n) x 15%
= 0.15 - 0.15n

Total proportion of small-sized artworks :
0.05n + (0.15 - 0.15n) = 10%
0.15 - 0.1n = 0.1
0.1n = 0.05
n = 0.5

Hence, proportion of artworks about fruit = 0.5

2.
a : customer selects stylist A
b : customer selects stylist B
c : customer selects stylist C

A : customer served by stylist A
B : customer served by stylist B
C : customer served by stylist C

P(1 customer served by favorite stylist)
= P(aA) + P(bB) + P(cC)
= 0.35 x 0.8 + 0.25 x 0.65 + 0.4 x 0.3
= 0.28 + 0.1625 + 0.12
= 0.5625

P(1 customer not served by favorite stylist)
= 1 - 0.5625
= 0.4375

P(only 1 customer served by favorite stylist among 3 customers)
= 3C1 x 0.5625 x 0.4375
= 3 x 0.5625 x 0.4375²
= 0.322998046875
≈ 0.32

2013-02-09 10:21:42 補充：
The last 4th line should be :
" = 3C1 x 0.5625 x 0.4375^2 "

參考: fooks, fooks

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