To solve this problem, we will use the principle of inclusion-exclusion and set notation to break down the information given in the survey. Let's define the sets:

• $G$: Set of adults who played golf
• $S$: Set of adults who skied
• $T$: Set of adults who played tennis

Given Information:

1. $|G| = 190$ (total who played golf)
2. $|S| = 200$ (total who skied)
3. $|T| = 95$ (total who played tennis)
4. $|G \cap \overline{S} \cap \overline{T}| = 100$ (played golf but did not ski or play tennis)
5. $|S \cap \overline{G} \cap \overline{T}| = 120$ (skied but did not play golf or tennis)
6. $|G \cap S \cap \overline{T}| = 30$ (played golf and skied but did not play tennis)
7. $|G \cap S \cap T| = 40$ (played all three)

a. How many played golf and tennis but did not ski?

We need to find $|G \cap T \cap \overline{S}|$, which is the number of adults who played both golf and tennis but did not ski.

The formula for the total number of people who played golf can be broken down as:

$|G| = |G \cap \overline{S} \cap \overline{T}| + |G \cap S \cap \overline{T}| + |G \cap \overline{S} \cap T| + |G \cap S \cap T|$

We know:

• $|G \cap \overline{S} \cap \overline{T}| = 100$
• $|G \cap S \cap \overline{T}| = 30$
• $|G \cap S \cap T| = 40$

Substitute these values:

$190 = 100 + 30 + |G \cap \overline{S} \cap T| + 40$

Simplify to find $|G \cap \overline{S} \cap T|$:

$190 = 170 + |G \cap \overline{S} \cap T|$

$|G \cap \overline{S} \cap T| = 20$

So, 20 adults played golf and tennis but did not ski.

b. How many played tennis but did not play golf or ski?

We need to find $|T \cap \overline{G} \cap \overline{S}|$, which is the number of adults who played tennis but did not play golf or ski.

The formula for the total number of people who played tennis can be broken down as:

$|T| = |T \cap \overline{G} \cap \overline{S}| + |G \cap \overline{S} \cap T| + |S \cap \overline{G} \cap T| + |G \cap S \cap T|$

We know:

• $|G \cap \overline{S} \cap T| = 20$
• $|G \cap S \cap T| = 40$

To find $|S \cap \overline{G} \cap T|$, we apply the principle:

$|S \cap \overline{G} \cap T| = |S \cap T| - |G \cap S \cap T| = |S \cap T| - 40$

However, we first find $|T \cap \overline{G} \cap S| = 95$

Total adults have golf, skiing, or tennis,

Now, skii, tennis, etc.

Finally,

Participants.

Feel free back.