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Sagot :
Let's carefully analyze the problem step by step:
### Part 1: Calculating the Probability of Selecting Coffee or Cookies
First, let's determine the total number of gift baskets in stock:
- Coffee containing cookies: 21
- Coffee containing mugs: 23
- Coffee containing candy: 17
- Tea containing cookies: 22
- Tea containing mugs: 17
- Tea containing candy: 22
Summing all these, we get the total number of gift baskets:
[tex]\[ \text{Total gift baskets} = 21 + 23 + 17 + 22 + 17 + 22 = 122 \][/tex]
Next, we need to find the total number of baskets that contain either coffee or cookies:
- Total baskets with coffee:
- Coffee with cookies: 21
- Coffee with mugs: 23
- Coffee with candy: 17
[tex]\[ \text{Total coffee baskets} = 21 + 23 + 17 = 61 \][/tex]
- Total baskets with cookies:
- Coffee with cookies: 21
- Tea with cookies: 22
[tex]\[ \text{Total cookies baskets} = 21 + 22 = 43 \][/tex]
However, since the baskets containing coffee and cookies are counted twice in both coffee and cookies categories, we subtract those overlapping baskets:
[tex]\[ \text{Total baskets with coffee or cookies} = 61 + 43 - 21 = 83 \][/tex]
Finally, the probability of a basket containing coffee or cookies is:
[tex]\[ P(\text{coffee or cookies}) = \frac{\text{Number of coffee or cookies baskets}}{\text{Total number of baskets}} = \frac{83}{122} \approx 0.680 \][/tex]
So, the probability rounded to 3 decimal places is 0.680.
### Part 2: Calculating the Probability of Tea Given That It Contains Mugs
Now, we need to find the probability that a basket contains tea, given it contains mugs:
First, find the total number of baskets containing mugs:
- Coffee with mugs: 23
- Tea with mugs: 17
[tex]\[ \text{Total baskets with mugs} = 23 + 17 = 40 \][/tex]
Next, find the number of baskets containing tea and mugs:
- Tea with mugs: 17
The probability of a basket containing tea given that it contains mugs is:
[tex]\[ P(\text{tea | mugs}) = \frac{\text{Number of tea baskets with mugs}}{\text{Total number of baskets with mugs}} = \frac{17}{40} \approx 0.425 \][/tex]
So, the probability rounded to 3 decimal places is 0.425.
### Summary of Answers:
(a) The probability of selecting a gift basket that contains coffee or cookies is [tex]\( P(\text{coffee or cookies}) \approx 0.680 \)[/tex].
(b) The probability of selecting a gift basket that contains tea, given it contains mugs is [tex]\( P(\text{tea | mugs}) \approx 0.425 \)[/tex].
### Part 1: Calculating the Probability of Selecting Coffee or Cookies
First, let's determine the total number of gift baskets in stock:
- Coffee containing cookies: 21
- Coffee containing mugs: 23
- Coffee containing candy: 17
- Tea containing cookies: 22
- Tea containing mugs: 17
- Tea containing candy: 22
Summing all these, we get the total number of gift baskets:
[tex]\[ \text{Total gift baskets} = 21 + 23 + 17 + 22 + 17 + 22 = 122 \][/tex]
Next, we need to find the total number of baskets that contain either coffee or cookies:
- Total baskets with coffee:
- Coffee with cookies: 21
- Coffee with mugs: 23
- Coffee with candy: 17
[tex]\[ \text{Total coffee baskets} = 21 + 23 + 17 = 61 \][/tex]
- Total baskets with cookies:
- Coffee with cookies: 21
- Tea with cookies: 22
[tex]\[ \text{Total cookies baskets} = 21 + 22 = 43 \][/tex]
However, since the baskets containing coffee and cookies are counted twice in both coffee and cookies categories, we subtract those overlapping baskets:
[tex]\[ \text{Total baskets with coffee or cookies} = 61 + 43 - 21 = 83 \][/tex]
Finally, the probability of a basket containing coffee or cookies is:
[tex]\[ P(\text{coffee or cookies}) = \frac{\text{Number of coffee or cookies baskets}}{\text{Total number of baskets}} = \frac{83}{122} \approx 0.680 \][/tex]
So, the probability rounded to 3 decimal places is 0.680.
### Part 2: Calculating the Probability of Tea Given That It Contains Mugs
Now, we need to find the probability that a basket contains tea, given it contains mugs:
First, find the total number of baskets containing mugs:
- Coffee with mugs: 23
- Tea with mugs: 17
[tex]\[ \text{Total baskets with mugs} = 23 + 17 = 40 \][/tex]
Next, find the number of baskets containing tea and mugs:
- Tea with mugs: 17
The probability of a basket containing tea given that it contains mugs is:
[tex]\[ P(\text{tea | mugs}) = \frac{\text{Number of tea baskets with mugs}}{\text{Total number of baskets with mugs}} = \frac{17}{40} \approx 0.425 \][/tex]
So, the probability rounded to 3 decimal places is 0.425.
### Summary of Answers:
(a) The probability of selecting a gift basket that contains coffee or cookies is [tex]\( P(\text{coffee or cookies}) \approx 0.680 \)[/tex].
(b) The probability of selecting a gift basket that contains tea, given it contains mugs is [tex]\( P(\text{tea | mugs}) \approx 0.425 \)[/tex].
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