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 5 times and lost 5 times.

1. Car
2. Car
3. Goat
4. Car
5. Car
6. Goat
7. Goat
8. Goat
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 319 times and lost 681 times.

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