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. Goat
8. Car
9. Car
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 341 times and lost 659 times.

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