Question

Before introducing a new toy to the market in time for Christmas, a toy manufacture sends prototypes to a panel of children whose responses are coded as strongly favorable, weakly favorable, weakly unfavorable and strongly unfavorable. Historically, 70% of all new toys introduced by the manufacturer have been successful on the market, the remainder proving unsuccessful. Similar panels have examined 1,000 toys in the past. The accompanying table shows their reactions and the resultant market performance of the toys

		Panel Reaction
		Strongly Favorable	Weakly Favorable	Weakly Unfavorable	Strongly Unfavorable
Market Performance	Successful	160	220	120	100
	Unsuccessful	20	60	100	220

1. 1. If the panel reaction is strongly favorable, what is the probability that the toy will be successful?
2. 2. If the panel reaction is strongly unfavorable, what is the probability that the toy will be unsuccessful?
3. 3. If the panel reaction is weakly favorable or better, what is the probability that the toy will be successful?
4. 4. If the toy turns out to be successful, what is the probability that the panel reaction will have been weakly unfavorable or worse?

Answer 1

Let S =Successful; S^c =Unsuccessful; Sf =Strongly Favorable; Wf =Weakly Favorable; Wu =Weakly Unfavorable; Su =Strongly Unfavorable; P =Probability

Given: P(S) =70% =0.70

1.

P(S | Sf) =P(S Sf)/P(Sf) =(160/1000)/(180/1000) =160/180 =8/9 =0.8889

2.

P(S^c | Su) =P(S^c Su)/P(Su) =(220/1000)/(320/1000) =220/320 =11/16 =0.6875

3.

P(S | (Wf or Sf)) =P(S (Wf or Sf))/P(Wf or Sf) =[(160+220)/1000]/[(160+220+20+60)/1000] =380/460 =19/23 =0.8261

4.

P((Wu or Su) | S) =P((Wu or Su) S)/P(S) =[(120+100)/1000]/0.70 =0.220/0.70 =0.3143