A science teacher presented a balloon to her class, and posed the question, "If I keep this balloon in the freezer overnight, what will occur?" She followed up with, "How certain are you of your answer?" The teacher recorded her students' responses.

Question

Shrink Expand
Certain 4 5
Uncertain 6 4
What is the probability that a randomly selected student was certain and believes the balloon will expand?
Simplify any fractions.

Answer 1

To find the probability that a randomly selected student was certain and believes the balloon will expand, we need to determine the number of students in that category and the total number of students.

From the table:

• The number of students who are certain that the balloon will expand is 5.
• The total number of students can be calculated by adding up all the responses:

\[ \text{Total students} = (4 \text{ certain, shrink}) + (5 \text{ certain, expand}) + (6 \text{ uncertain, shrink}) + (4 \text{ uncertain, expand}) = 4 + 5 + 6 + 4 = 19 \]

Now, we have:

• Number of certain students who believe the balloon will expand = 5
• Total number of students = 19

Thus, the probability \( P \) that a randomly selected student was certain and believes the balloon will expand is given by:

\[ P(\text{certain and expand}) = \frac{\text{Number of certain, expand}}{\text{Total students}} = \frac{5}{19} \]

So the probability that a randomly selected student was certain and believes the balloon will expand is

\[ \frac{5}{19} \]