Gift Baskets

The Gift Basket Store has the following premade gift baskets in stock containing the following combinations:

\begin{tabular}{lccc}
& Cookies & Mugs & Candy \\
\hline
Coffee & 21 & 23 & 17 \\
Tea & 22 & 17 & 22
\end{tabular}

Choose 1 basket at random. Find the probability that it contains the following combinations. Enter your answers as fractions or as decimals rounded to 3 decimal places.

(a) Coffee or cookies:
[tex]\[ P(\text{coffee or cookies}) = \frac{83}{122} \][/tex]

(b) Tea, given that it contains mugs:
[tex]\[ P(\text{tea, given that it contains mugs}) = \square \][/tex]

Part 1 of 3



Answer :

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].