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. Car
4. Goat
5. Goat
6. Car
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 327 times and lost 673 times.

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