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 3 times and lost 7 times.

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

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 342 times and lost 658 times.

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