Probability of an Event

Numbers from Zero to One Describe Uncertainty

Will it rain today? Will a team win its next match? These outcomes are uncertain. Probability expresses how likely each outcome is with a number from $0$ to $1$.

Probability measures how likely an event is. Its value lies between $0$ and $1$:

• Probability $0$ means the event is impossible
• Probability $1$ means the event is certain
• Probability $0.5$ means the event and its complement are equally likely

Probability does not predict one outcome with certainty. It quantifies uncertainty so repeated outcomes can be compared and decisions can state their uncertainty explicitly.

Sample Space and Events

Two ideas are needed before we calculate a probability:

The sample space (S) is the set of every possible outcome of an experiment. When two dice are rolled, for example, each outcome is an ordered pair of face values.

An event (A) is a subset of the sample space containing the outcomes we want to examine.

Concrete example: for two coin tosses, let $H$ denote heads and $T$ tails. The sample space is $\{HH, HT, TH, TT\}$. The event “at least one tail” is $A = \{HT, TH, TT\}$.

Probability Formula

To calculate the probability of an event, we use the classical probability formula:

The quantities in this formula are:

• $P(A)$ = probability of event $A$ occurring
• $n(A)$ = number of favorable outcomes (members of event $A$)
• $n(S)$ = number of all possible outcomes (members of sample space)

Condition: This counting formula applies only when all outcomes in the sample space are equally likely.

Every event probability satisfies:

• $0 \leq P(A) \leq 1$ (probability is always between $0$ and $1$)
• $P(A) + P(A') = 1$, where $A'$ is the complement of event $A$

Complement: The complement of $A$ contains every outcome in the sample space that is not in $A$. If $A$ is “rolling an even number,” then $A'$ is “rolling an odd number.”

Probability That Two Dice Sum to Nine

Analysis of Rolling Two Dice:

Two fair dice produce $6 \times 6 = 36$ equally likely ordered pairs. Find the probability that their sum is $9$.

Systematic solution:

1. Identify all ways to get sum $9$:

   • First die $3$, second die $6$: $(3,6)$
   • First die $4$, second die $5$: $(4,5)$
   • First die $5$, second die $4$: $(5,4)$
   • First die $6$, second die $3$: $(6,3)$

2. Count the favorable outcomes: $n(A) = 4$

3. Determine probability:

   $$P(\text{sum} = 9) = \frac{4}{36} = \frac{1}{9} \approx 0.111$$

Probability-Based Marketing Strategy:

A beverage company inserts prize coupons into some milk boxes. Historical data estimate the probability that a purchased box contains a prize as $\frac{3}{32}$.

Practical interpretation: Over many purchases, about $3$ of every $32$ purchased boxes are expected to contain a prize. The estimate can help the company:

• Plan promotional budgets
• Estimate consumer response
• Determine sales targets

Comparing Dice Sums:

For two fair dice, a sum of $7$ has more favorable ordered pairs than any other sum:

There are $6$ ways to get a sum of $7$: $(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)$.

This is the largest probability among the possible sums. It does not guarantee that the next roll will be $7$. It describes the long-run model.

Practice Problems

Solve each problem before reading the worked solution. Define the sample space and event explicitly before substituting numbers into a formula. Both problems count equally likely outcomes, so write the counts first and divide them on the last step.

1. In rolling two dice, determine the probability of getting a sum that is an even number.

2. A box contains $5 \text{ red balls}$, $3 \text{ blue balls}$, and $2 \text{ green balls}$. If one ball is drawn randomly, determine the probability of getting a ball that is not red.

3. Three fair coins are tossed independently. Determine the probability of getting exactly two tails.

4. A quality inspection finds exactly $50$ defective items in a batch of $1000 \text{ units}$. If one item is selected uniformly at random, what is the probability that it is not defective?

Worked Solutions

1. Answer: $\frac{1}{2}$

   Systematic solution steps:

   Sample space for rolling two dice: $n(S) = 36$

   Identify all even sums and ways to obtain them:

   • Sum $2$: $(1,1)$ → $1$ way
   • Sum $4$: $(1,3), (2,2), (3,1)$ → $3 \text{ ways}$
   • Sum $6$: $(1,5), (2,4), (3,3), (4,2), (5,1)$ → $5 \text{ ways}$
   • Sum $8$: $(2,6), (3,5), (4,4), (5,3), (6,2)$ → $5 \text{ ways}$
   • Sum $10$: $(4,6), (5,5), (6,4)$ → $3 \text{ ways}$
   • Sum $12$: $(6,6)$ → $1$ way

   Total favorable outcomes: $n(A) = 1 + 3 + 5 + 5 + 3 + 1 = 18$

   Probability calculation:

   $$P(\text{even}) = \frac{18}{36} = \frac{1}{2} = 0.5$$

2. Answer: $\frac{1}{2}$

   Solution steps:

   Count total balls: $5 + 3 + 2 = 10$ balls

   Identify balls that are not red: $3 \text{ blue} + 2 \text{ green} = 5$ balls

   Probability calculation:

   $$P(\text{not red}) = \frac{5}{10} = \frac{1}{2} = 0.5$$

3. Answer: $\frac{3}{8}$

   Solution steps:

   Sample space for three coin tosses:

   $$\left\{\begin{gathered} HHH, HHT, HTH, \\ HTT, THH, THT, \\ TTH, TTT \end{gathered}\right\}$$

   Total possibilities: $n(S) = 2^3 = 8$

   Event exactly two tails: $\{HTT, THT, TTH\}$

   Number of favorable outcomes: $n(A) = 3$

   Probability calculation:

   $$P(\text{exactly 2 tails}) = \frac{3}{8} = 0.375$$

4. Answer: $\frac{19}{20}$ or $95\%$

   Solution steps:

   Total products: $1000 \text{ units}$

   Defective products: $5\% \times 1000 = 50 \text{ units}$

   Non-defective products: $1000 - 50 = 950 \text{ units}$

   Probability calculation:

   $$P(\text{not defective}) = \frac{950}{1000} = \frac{19}{20} = 0.95 = 95\%$$

   Interpretation: Under uniform random selection from this inspected batch, an item is non-defective with probability $95\%$. This does not guarantee the result of one selection.