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.

Survival of poisoned silkworm larvae (seconds)
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.

\[ \text{estimate of } P(survival \text{ }time \ge 300) = \frac {12} {80} = 0.15 \]