From Group Data to Useful Probabilities: The Relevance of Actuarial Risk Assessment in Individual Instances

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Figure 1.
Ticket of the type bought and sold by Mary and Kathy as they bet on outcomes of penny drawings.

Table 1

Results and Inferences About π_y, the Proportion of Denver Pennies in Jar y, After n Draws (With Replacement) From Each Jar in the Penny Collection

y	r	n	π̂_y and 95% Credible Intervals
y	r	n	Mary's Estimates		Kathy's Estimates^*
1	0	11	0.042	[0.000–0.200]	0.039	[0.022–0.063]
2	6	71	0.090	[0.036–0.166]	0.068	[0.044–0.010]
3	12	101	0.123	[0.067–0.192]	0.118	[0.086–0.155]
4	19	111	0.174	[0.110–0.249]	0.197	[0.159–0.237]
5	41	116	0.355	[0.271–0.443]	0.310	[0.270–0.352]
6	42	96	0.438	[0.341–0.537]	0.453	[0.401–0.505]
7	41	74	0.553	[0.441–0.663]	0.603	[0.535–0.670]
8	22	29	0.750	[0.584–0.885]	0.736	[0.656–0.806]
9	9	9	0.950	[0.762–1.000]	0.835	[0.761–0.895]

r, number of Denver Pennies; y, number of years before Dorothy's death.
↵* Inferences are based on 25,000 iterations of two WinBUGS chains that converged rapidly (after 100 updates). The first 5,000 values from each chain were discarded, and inferences were made on the final 20,000 values in each chain, thinned at an interval of 20.