In the class of 30 children, there are 20 who can swim and 25 who can ride a bike. There are no chil who cannot swim nor ride a bike.

Question

Draw a Venn diagram and find the probability that a child selected at random:
(a) Can swim and ride a bike
(b) Cannot swim but can ride a bike
(c) Can swim but cannot ride a bike

Answer 1

To solve the problem, we will analyze the information given about the class of 30 children using a Venn diagram, and then calculate the probabilities required.

1. Understanding the Information:

   • Total children: 30
   • Children who can swim: 20
   • Children who can ride a bike: 25
   • No children can be described as neither swimming nor riding a bike.

2. Setting Up the Venn Diagram: We can denote:

   • \( S \): the set of children who can swim
   • \( B \): the set of children who can ride a bike

   To find the overlap (children who can both swim and ride a bike), we need to apply the principle of inclusion-exclusion:

   \[ |S \cup B| = |S| + |B| - |S \cap B| \]

   Here, \( |S \cup B| \) is the total number of unique children (which is 30).

   Plugging in the numbers we have: \[ 30 = 20 + 25 - |S \cap B| \] \[ 30 = 45 - |S \cap B| \] \[ |S \cap B| = 45 - 30 = 15 \]

   Thus, there are 15 children who can both swim and ride a bike.

3. Finding the Exclusive Sets:

   • Children who can only swim (but not ride a bike): \[ |S| - |S \cap B| = 20 - 15 = 5 \]
   • Children who can only ride a bike (but not swim): \[ |B| - |S \cap B| = 25 - 15 = 10 \]

4. Completing the Venn Diagram:

   • Children who can swim only: 5
   • Children who can ride a bike only: 10
   • Children who can do both: 15

   To summarize:

   • Only swim: 5
   • Only bike: 10
   • Both: 15

5. Calculating Probabilities: Now, let’s compute the probabilities for each part:

(a) Probability that a randomly selected child can swim and ride a bike: \[ P(\text{swim and bike}) = \frac{|S \cap B|}{\text{Total children}} = \frac{15}{30} = \frac{1}{2} \]

(b) Probability that a randomly selected child cannot swim but can ride a bike:

• Since all children can swim or ride a bike, there is no child who cannot swim but can ride a bike. \[ P(\text{not swim but bike}) = 0 \]

(c) Probability that a randomly selected child can swim but cannot ride a bike: \[ P(\text{swim but not bike}) = \frac{|S| - |S \cap B|}{\text{Total children}} = \frac{5}{30} = \frac{1}{6} \]

Summary of Results:

• (a) Probability that a child can swim and ride a bike: \( \frac{1}{2} \)
• (b) Probability that a child cannot swim but can ride a bike: \( 0 \)
• (c) Probability that a child can swim but cannot ride a bike: \( \frac{1}{6} \)