Unknown probabilities
We have now given a definition of the probability of an event as the long-term proportion of times that it occurs when the experiment is repeated. Unfortunately, although this explains the concept of the event's probability, it does not provide a numerical value in practice, since we cannot wait until the experiment has been repeated an infinite number of times. When dealing with experiments that do not have equally-likely outcomes, we can rarely find the exact probabilities of events.
However although we cannot repeat the experiment an infinite number of times, the proportion of times that the event occurs in a large number of repetitions will give an approximate value for the probability — an estimate of its value.
Survival of silkworm larvae
Silkworm larvae weighing between 0.41 and 0.45 grams were given 0.10 mg of sodium arsenate per gram of body weight. Their survival times in seconds are given below.
270 254 293 244 293 261 285 330 284 274 |
307 235 215 292 309 267 275 298 241 254 |
256 275 226 287 280 339 294 298 283 366 |
300 310 280 240 291 286 230 285 218 279 |
280 286 345 289 210 282 260 228 243 259 |
285 275 280 296 283 248 314 258 215 299 |
240 241 236 255 267 271 253 271 233 260 |
273 233 271 267 258 319 310 302 260 251 |
The probability that a single silkworm larva survives at least 300 seconds is unknown. However this experiment has been repeated 80 times with different larvae, so we can estimate this probability from the proportion of survival times that were at least 300 seconds in these 80 repetitions.
\[ \text{estimate of } P(survival \text{ }time \ge 300) = \frac {12} {80} = 0.15 \]