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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Car
6. Goat
7. Goat
8. Car
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 341 times and lost 659 times.

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