A binary outcome is a trial result with exactly two possible states: success (S) or failure (F). In probability modeling, each state receives a probability, such as π for success and 1 − π for failure. section 5.9.3
Repeated binary outcomes form the basis of the binomial distribution. For independent trials, the probability of a particular sequence is the product of its success and failure probabilities; probabilities for equivalent sequences are then summed. section 5.9.3
Celebrity-photo trials
If 20% of photos contain a celebrity, then π = 0.2 and failure has probability 0.8. The sequence {S, S, F} has probability 0.2 × 0.2 × 0.8 = 0.032; the three sequences containing two successes produce a total probability of 0.096. section 5.9.3
Table 5.8 All possible combinations of three trials
| Event Id |
Event |
Probability |
|---|---|---|
| 0 |
{F, F, F} |
(1−π) × (1−π) × (1−π) = 0.8 × 0.8 × 0.8 |
| 1 |
{F, F, S} |
(1−π) × (1−π) × π = 0.8 × 0.8 × 0.2 |
| 2 |
{F, S, F} |
(1−π) × π × (1−π) = 0.8 × 0.2 × 0.8 |
| 3 |
{F, S, S} |
(1−π) × π × π = 0.8 × 0.2 × 0.2 |
| 4 |
{S, F, F} |
π × (1−π) × (1−π) = 0.2 × 0.8 × 0.8 |
| 5 |
{S, F, S} |
π × (1−π) × π = 0.2 × 0.8 × 0.2 |
| 6 |
{S, S, F} |
π × π × (1−π) = 0.2 × 0.2 × 0.8 |
| 7 |
{S, S, S} |
π × π × π = 0.2 × 0.2 × 0.2 |
Enumerates all eight success–failure sequences for three trials and gives each sequence's probability. section 5.9.3
