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 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
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 317 times and lost 683 times.

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

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 28 times and lost 72 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;
	}
}