Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 2 times and lost 8 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 319 times and lost 681 times.

1. Car
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Car
14. Goat
15. Goat
16. Car
17. Goat
18. Goat
19. Goat
20. Car
21. Goat
22. Car
23. Goat
24. Goat
25. Goat
26. Car
27. Car
28. Goat
29. Goat
30. Goat
31. Goat
32. Goat
33. Car
34. Car
35. Car
36. Goat
37. Goat
38. Goat
39. Goat
40. Goat
41. Car
42. Goat
43. Car
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Car
50. Goat
51. Car
52. Goat
53. Goat
54. Car
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Goat
62. Car
63. Goat
64. Car
65. Goat
66. Car
67. Car
68. Goat
69. Goat
70. Goat
71. Goat
72. Car
73. Car
74. Car
75. Goat
76. Car
77. Goat
78. Goat
79. Goat
80. Car
81. Goat
82. Car
83. Goat
84. Goat
85. Car
86. Goat
87. Goat
88. Car
89. Goat
90. Car
91. Car
92. Car
93. Goat
94. Goat
95. Goat
96. Goat
97. Goat
98. Goat
99. Goat
100. Goat
101. Goat
102. Goat
103. Goat
104. Goat
105. Car
106. Goat
107. Car
108. Goat
109. Goat
110. Goat
111. Car
112. Goat
113. Goat
114. Car
115. Car
116. Goat
117. Goat
118. Car
119. Car
120. Goat
121. Goat
122. Car
123. Goat
124. Car
125. Goat
126. Car
127. Goat
128. Goat
129. Car
130. Car
131. Goat
132. Car
133. Goat
134. Goat
135. Goat
136. Goat
137. Goat
138. Car
139. Goat
140. Car
141. Goat
142. Goat
143. Car
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Car
153. Goat
154. Goat
155. Car
156. Goat
157. Goat
158. Car
159. Goat
160. Goat
161. Goat
162. Car
163. Car
164. Goat
165. Goat
166. Goat
167. Car
168. Goat
169. Goat
170. Car
171. Car
172. Goat
173. Goat
174. Car
175. Goat
176. Goat
177. Car
178. Goat
179. Car
180. Goat
181. Goat
182. Goat
183. Goat
184. Goat
185. Goat
186. Car
187. Goat
188. Car
189. Goat
190. Car
191. Car
192. Goat
193. Goat
194. Car
195. Goat
196. Goat
197. Goat
198. Goat
199. Goat
200. Goat
201. Car
202. Car
203. Goat
204. Goat
205. Car
206. Goat
207. Car
208. Goat
209. Goat
210. Goat
211. Car
212. Goat
213. Car
214. Goat
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Car
221. Car
222. Goat
223. Car
224. Car
225. Goat
226. Goat
227. Car
228. Goat
229. Car
230. Goat
231. Goat
232. Goat
233. Car
234. Goat
235. Car
236. Goat
237. Goat
238. Goat
239. Goat
240. Goat
241. Goat
242. Goat
243. Goat
244. Car
245. Goat
246. Goat
247. Goat
248. Goat
249. Car
250. Goat
251. Goat
252. Goat
253. Car
254. Car
255. Car
256. Car
257. Goat
258. Goat
259. Goat
260. Goat
261. Goat
262. Car
263. Goat
264. Goat
265. Goat
266. Goat
267. Goat
268. Car
269. Car
270. Goat
271. Goat
272. Car
273. Goat
274. Car
275. Goat
276. Goat
277. Car
278. Goat
279. Goat
280. Car
281. Goat
282. Goat
283. Goat
284. Goat
285. Goat
286. Car
287. Car
288. Goat
289. Car
290. Goat
291. Car
292. Car
293. Goat
294. Goat
295. Goat
296. Goat
297. Car
298. Car
299. Car
300. Car
301. Car
302. Goat
303. Goat
304. Goat
305. Car
306. Goat
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Car
313. Goat
314. Goat
315. Goat
316. Car
317. Car
318. Car
319. Car
320. Goat
321. Car
322. Car
323. Goat
324. Goat
325. Goat
326. Car
327. Goat
328. Goat
329. Goat
330. Goat
331. Car
332. Goat
333. Car
334. Goat
335. Goat
336. Car
337. Goat
338. Goat
339. Goat
340. Goat
341. Goat
342. Car
343. Car
344. Car
345. Goat
346. Goat
347. Goat
348. Goat
349. Car
350. Goat
351. Goat
352. Goat
353. Goat
354. Car
355. Goat
356. Goat
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Goat
363. Goat
364. Goat
365. Goat
366. Goat
367. Goat
368. Car
369. Goat
370. Car
371. Goat
372. Goat
373. Goat
374. Goat
375. Goat
376. Car
377. Car
378. Goat
379. Goat
380. Goat
381. Goat
382. Goat
383. Goat
384. Goat
385. Goat
386. Car
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Goat
393. Goat
394. Car
395. Goat
396. Goat
397. Goat
398. Goat
399. Car
400. Goat
401. Car
402. Car
403. Goat
404. Car
405. Goat
406. Goat
407. Car
408. Goat
409. Car
410. Car
411. Car
412. Goat
413. Goat
414. Goat
415. Goat
416. Goat
417. Goat
418. Goat
419. Goat
420. Car
421. Goat
422. Car
423. Goat
424. Goat
425. Car
426. Goat
427. Car
428. Car
429. Car
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Goat
436. Car
437. Car
438. Car
439. Car
440. Goat
441. Car
442. Car
443. Car
444. Car
445. Goat
446. Goat
447. Goat
448. Goat
449. Car
450. Goat
451. Goat
452. Car
453. Goat
454. Car
455. Car
456. Goat
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Car
463. Car
464. Goat
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Car
471. Car
472. Goat
473. Goat
474. Goat
475. Car
476. Goat
477. Goat
478. Goat
479. Goat
480. Goat
481. Car
482. Goat
483. Goat
484. Goat
485. Car
486. Goat
487. Goat
488. Car
489. Goat
490. Goat
491. Goat
492. Goat
493. Goat
494. Car
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Goat
502. Goat
503. Goat
504. Goat
505. Car
506. Car
507. Car
508. Goat
509. Car
510. Goat
511. Goat
512. Goat
513. Goat
514. Car
515. Goat
516. Goat
517. Goat
518. Goat
519. Goat
520. Car
521. Car
522. Goat
523. Goat
524. Goat
525. Car
526. Goat
527. Goat
528. Goat
529. Car
530. Car
531. Car
532. Goat
533. Car
534. Goat
535. Goat
536. Goat
537. Goat
538. Car
539. Goat
540. Goat
541. Goat
542. Goat
543. Goat
544. Goat
545. Car
546. Goat
547. Goat
548. Goat
549. Goat
550. Goat
551. Goat
552. Goat
553. Goat
554. Goat
555. Car
556. Goat
557. Goat
558. Car
559. Goat
560. Goat
561. Car
562. Car
563. Car
564. Goat
565. Goat
566. Car
567. Goat
568. Goat
569. Goat
570. Goat
571. Goat
572. Goat
573. Goat
574. Goat
575. Goat
576. Car
577. Goat
578. Car
579. Goat
580. Car
581. Goat
582. Car
583. Goat
584. Car
585. Goat
586. Car
587. Car
588. Goat
589. Goat
590. Goat
591. Goat
592. Goat
593. Goat
594. Goat
595. Car
596. Goat
597. Goat
598. Goat
599. Car
600. Goat
601. Goat
602. Goat
603. Goat
604. Goat
605. Goat
606. Goat
607. Car
608. Car
609. Goat
610. Goat
611. Goat
612. Goat
613. Car
614. Goat
615. Goat
616. Goat
617. Car
618. Car
619. Car
620. Car
621. Goat
622. Goat
623. Car
624. Car
625. Car
626. Car
627. Goat
628. Car
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Car
635. Goat
636. Car
637. Car
638. Goat
639. Goat
640. Goat
641. Car
642. Car
643. Goat
644. Goat
645. Car
646. Goat
647. Goat
648. Goat
649. Goat
650. Goat
651. Car
652. Goat
653. Car
654. Goat
655. Goat
656. Goat
657. Goat
658. Goat
659. Goat
660. Goat
661. Car
662. Goat
663. Goat
664. Goat
665. Goat
666. Goat
667. Goat
668. Car
669. Car
670. Goat
671. Goat
672. Car
673. Goat
674. Goat
675. Car
676. Goat
677. Goat
678. Car
679. Car
680. Goat
681. Goat
682. Goat
683. Car
684. Goat
685. Goat
686. Goat
687. Goat
688. Goat
689. Goat
690. Goat
691. Goat
692. Car
693. Goat
694. Goat
695. Car
696. Car
697. Car
698. Goat
699. Goat
700. Goat
701. Goat
702. Goat
703. Goat
704. Goat
705. Car
706. Car
707. Goat
708. Goat
709. Car
710. Goat
711. Goat
712. Goat
713. Goat
714. Car
715. Goat
716. Goat
717. Car
718. Car
719. Goat
720. Goat
721. Goat
722. Goat
723. Car
724. Goat
725. Car
726. Car
727. Goat
728. Car
729. Car
730. Goat
731. Car
732. Goat
733. Car
734. Goat
735. Car
736. Car
737. Goat
738. Goat
739. Goat
740. Car
741. Goat
742. Car
743. Goat
744. Car
745. Goat
746. Car
747. Goat
748. Car
749. Car
750. Goat
751. Goat
752. Goat
753. Car
754. Goat
755. Car
756. Goat
757. Goat
758. Goat
759. Car
760. Goat
761. Goat
762. Goat
763. Goat
764. Car
765. Car
766. Car
767. Goat
768. Car
769. Car
770. Goat
771. Car
772. Car
773. Car
774. Goat
775. Goat
776. Goat
777. Goat
778. Goat
779. Car
780. Goat
781. Goat
782. Goat
783. Goat
784. Goat
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Goat
792. Goat
793. Car
794. Car
795. Goat
796. Car
797. Goat
798. Goat
799. Goat
800. Car
801. Car
802. Goat
803. Goat
804. Goat
805. Goat
806. Car
807. Goat
808. Goat
809. Goat
810. Goat
811. Goat
812. Car
813. Car
814. Car
815. Car
816. Goat
817. Goat
818. Goat
819. Car
820. Goat
821. Goat
822. Goat
823. Goat
824. Goat
825. Goat
826. Car
827. Goat
828. Goat
829. Goat
830. Car
831. Goat
832. Goat
833. Goat
834. Car
835. Car
836. Goat
837. Goat
838. Goat
839. Car
840. Goat
841. Car
842. Goat
843. Goat
844. Goat
845. Goat
846. Goat
847. Goat
848. Goat
849. Goat
850. Goat
851. Car
852. Car
853. Car
854. Goat
855. Goat
856. Goat
857. Car
858. Car
859. Goat
860. Goat
861. Goat
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Car
871. Car
872. Car
873. Goat
874. Car
875. Goat
876. Goat
877. Goat
878. Car
879. Goat
880. Goat
881. Goat
882. Car
883. Goat
884. Goat
885. Car
886. Car
887. Goat
888. Goat
889. Car
890. Goat
891. Car
892. Goat
893. Goat
894. Goat
895. Goat
896. Goat
897. Car
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Goat
904. Goat
905. Car
906. Goat
907. Car
908. Goat
909. Goat
910. Goat
911. Car
912. Car
913. Car
914. Goat
915. Goat
916. Goat
917. Goat
918. Car
919. Goat
920. Goat
921. Goat
922. Goat
923. Goat
924. Goat
925. Goat
926. Goat
927. Goat
928. Goat
929. Goat
930. Goat
931. Car
932. Goat
933. Car
934. Goat
935. Goat
936. Goat
937. Goat
938. Goat
939. Goat
940. Goat
941. Car
942. Car
943. Car
944. Goat
945. Goat
946. Goat
947. Goat
948. Car
949. Goat
950. Goat
951. Goat
952. Goat
953. Car
954. Car
955. Car
956. Goat
957. Goat
958. Goat
959. Goat
960. Goat
961. Car
962. Goat
963. Goat
964. Car
965. Car
966. Goat
967. Car
968. Car
969. Goat
970. Car
971. Car
972. Goat
973. Goat
974. Car
975. Car
976. Goat
977. Goat
978. Car
979. Car
980. Goat
981. Car
982. Goat
983. Car
984. Goat
985. Goat
986. Goat
987. Goat
988. Goat
989. Goat
990. Goat
991. Goat
992. Car
993. Goat
994. Goat
995. Car
996. Goat
997. Goat
998. Goat
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 98 times and lost 902 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 32 times and lost 68 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}